Arithmetic on Elliptic Curves with Complex Multiplication

•

Lecture Notes In Mathematics

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Arithmetic on Elliptic Curves with Complex Multiplication

II,

With an Appendix by B. Mazur

~

Springer-Verlag 5 9 <j c; Berlin Heidelberg New York 1980

=

Table of Contents

o.

Introduction

1.

Acknowledgements

1 2

2.

Notation

3

Chapter 1:

3. 4. i

7.

8. Chapter 2:

9. 10. 11.

12. 14.

4:

17.

18. 19. 20.

Chapter 5:

21. 22. 23.

24. Appendix

, ,

I f

t

20

29 32

Local arithmetic

34 38 42

Global arithmetic

15. Restriction of Scalars 16.

12

14 17

23

A classification over F A rational p-isogeny Local invariants and global torsion

13.

4 8

A classification

Curves over H . Descended curves Ill-curves.

Chapter 3:

Chapter

The theory of complex multiplication Elliptic curves Elliptic curves over II: and E The analytic theory of complex multiplication Elliptic curves over p-adic fields ~adic Galois representations. The arithmetic theory of complex multiplication

5. 6.

,

and Conventions

The Ill-rank The first descent . A factorization of the L-series The sign in the functional equation tQ..... curves and modular forms . The Ill-curve

45 49 53

57 60 64

A(p )

67

Periods The rank of A(p) • Global models Computational examples

72

80

82

(by B. Mazur)

25.

The cohomology of the Fermat group scheme

87

26. 27.

Bibliography Index.

92 94

•

O.

Introduction.

K be an imaginary quadratic field, with ring of integers

Let number

h

and let

A be an elliptic curve over

Let

j (A)

of degree

h

be the modular invariant of

over

Sk~d

Then

j (A)

This fundamental result has its practical drawbacks. cannot be defined over

~

0,

is an algebraic integer

its conjugates generate the Hilbert class-field

~

class-

with complex multiplication by

~

A.

0

when the class-number of

H of

K.

For example, the curve

K is greater than one.

A

We can

often circumvent this problem by passing to the category of elliptic curves up to

isogeny.

of curves defined over their field of moduli all of their Galois conjugates

they

~

K is odd, one has a large supply

Specifically, when the discriminant of

defined over

Ql.

~

.!!.

F

=

Ill(j (A))

which are isogenous to

Arithmetically these curves behave as if

I call them (l-curves, and these notes are devoted to

their study 1

In Chapter 1 we recall some of the general theory of elliptic curves with complex. multiplication.

The treatment will be brief:

this sUbject have already appeared in print. curves' A over scend to

F

In Chapter 2 we classify elliptic

H with complex multiplication by

= Ill(j (A»

many excellent references on

0

We show which curves de-

and which descended curves are actually Ill-curves.

In Chapter 3, we study the arithmetic of descended curves at all completions of the field nant

-p.

F.

For simplicity, we restrict to the case where. K

In Chapter

4 we investigate the global arithmetic of Ill-curves.

5 is devoted to a detailed study of the ill-curve integers

!

Chapter

A(p) • with multiplication by the

0 of III( ,cp) and good reduction at all places of F

We end with a discussion of some questions which remain open.

I

has prime discrimi-

not dividing

p.

1.

Acknowledgements.

It is a pleasure to acknowledge the mathematical assistance I received from Joe Buhler, Pierre Deligne, Ken Kramer, Barry Mazur, Gilles Robert, David Rohrlich,

Jean-Pierre Serre, and Don Zagier. cellent job of typing.

,

t

I also wish to thank Lauri M. Hein for the ex-

Much more than thanks are due my family and friends --

Debby Gans, Ian Morrison, and Jane Reynolds -- who kept me distracted during the write-up.

Finally, I want to thank my teacher, John Tate, for all the inspiration

and support he gave me in the course of this work. M

r

fiI :J

no· f

Princeton, New Jersey

July, 1979

ga

2.

Notation and Conventions.

pm

Groups

G will always act on the left.

m~

action will be written either as

Lch,

sub-module of G-invariants.

e:x-

action of

a

If

and

M

On a homomorphism

u(m)

f:M

If

M is a G-module and

or

m t----+

um

E

G , this

G M denote the

We let

are G-modules, so is

N

a

Hom(M,N) :

the

is given by

--+ N

je ion

Rings will also act on the left. M r

=

{m EM: rm

=

If

B

A and

O}

fined over the field

If

M is an R-module and

we write

(fA

Ho~(A,B)

F , we let

and

gate homomorphism from

t

If

F

R

we let

are elliptic curves (or, more generally, abelian varieties) de-

F.

If

be the group of algebraic homomorphisms S

is any F-algebra, we let

note the abelian group of all S-rational points of F

g

denote the sub-module of "r-torsion."

$:A --+ B which are defined over

of

r

is a field,

If

a

for the conjugate varieties, and

(fB

crA to

F

A

A(S)

de-

is any automorphism a$

for the conju-

cr B .

denotes an algebraic closure of it.

We shall

alw~s

use

i

the isomorphism of local class field theory which takes a uniformizing parameter to

I

an arithmetic Frobenius in the Galois group.

•

I t

Chapter 1

3.

Elliptic curves 3.1.

The theory of complex multiplication

Deligne [6], Tate [29]).

(References:

An elliptic curve

singular curve of genus

A over the field

lover

F

F, furnished with a F-rational point

the theorem of Riemann-Roch, there exist functions on

is a complete, irreducible,

A which are regular outside of

0A'

x

and

y

of degree

0A 2

000-

By and

3

These functions, when suitably normalized,

Then we

satisfy an equation

'h

where the coefficients

Weierstrass model for

lie in

A.

Then

x

We call such an equation a generalized

F.

and y

generate the function field

F(A) ,

fine~

b

inva:""":B

and the above model is unique up to a change of coordinates of the form: 2

x' = u x + r in

where

u

is in F*

and

r,s,t

are in F.

Associated to the model (3.1.1) we

have the non-Zero differential of the first kind

This gives a basis for the F-vector space

HO(A,Ol).

Under a change of coordinates

£&)1

menta

t

variant

(3.1.2) We find

3.2.

Henc

=

U

-1

•

to •

Given a generalized Weierstrass model for

A over

F. define the ale-

3. beeD

5

(3.2.1)

= a 12

4

= a1a 3

6

= a 32

8

= b 2a 6

b b b

non-

+ 4a

2

b

c4

= b 22

c

6

= _b 23

= -b 2b 8

8b 3 4

2

+ 2a4

+ 4a

6

6

2

_ 24b

4

+ 36b b - 216b 2 4 6

27b

2 b b b 6 + 9 2 4 6

2 2 - a a a4 + a a - a4 2 3 1 3

3

Then we have the relation

The condition that

tJ." 0

is equivalent to the assertion that the curve de-

fined by (3.1.1) is non-singular.

This being the case, we may define the "modular

invariant It :

Under a change of coordinates (3.1.2) we find (3.2.4)

;es

Hence the quantities variant

3.3

j

= j (A)

If

c4' c6

and

tJ.

c4 •

= u 4c 4

c6 •

= u6c 6

tJ.'

= u 12tJ.

j

=j

I

depend only on the pair

depends only on the curve

A and

become isomorphic over

B are two curves ov"r

F.

. A.

F

If

with

a

is any automorphism of

j (A)

More generally, assume that

(A,.. ) , and the in-

F

= j (B)

, then

A and

F:

B

is a perfect field and

6

G = Gal(F!F)

let

Then there is a bijection between the pointed sets

B/F} +---+ ~(G,AutF-(A»)

{Isomorphism classes of with j(B) = j(A)

which takes If

A to the trivial class.

j(B) = j(A)

This bijection is constructed as follows.

we may choose an isomorphism

$:A ~ B

over

F.

The

H

assignment ~:G ---+ Autp(A)

a>---> $-1

is a continuous l-cocycle on phism class of

B =

B.

0

a$

G whose cohomology class depends only on the F-isomor-

Conversely, given such a cocycle

tIJ

one can construct a "twist"

A~ over F with j(B) = j(A) B(F) = ip

The isomorphism class of

3.4.

B

over

Since the curve

A

a(P) =

A(F)

E

F

~(a)

0

p} •

depends only on the cohomology class of

is isomorphic to its Jacobian over

the structure of an abelian variety.

The distinguished point

the identity in the algebraic group.

Any non-zero homomorphism

called an isogeny.

The (separable) degree of

corresponding field extension: The set law on

B.

Ho~(A.B)

The group

[F(A): $

0

En~(A)

0A

corresponds to

$:A --+ B

is

is the (separable) degree of the

F(B)]

forms a ring with multiplication given by the com-

For any

m

E ~

plication by m" in the lalgebraic group of

PA

F, it inherits

inherits the structure of an abelian group from the addition

position of homomorphisms'.

separable i f and only if

$

~.

m

A

is prime to· char (F)

IIllIY have separable degree either

ordinary, in the second case

we let

A

p

or

0

m be the endomorphism "multiA 2 This isogeny has degree m and is If

char (F)

=p

In the first case

, the isogeny A

is said to be

is said to be supersingular.

rt

' ~..•.

lC

7

If

$:A --+ B

defined over

F

is any isogen>' of degree

m, there is 'a dual isogeny

v-

$:B --+ A

with v~ 0

$ =

$

v~ =~

0

ffi

A

Hence the relation of isogeny is an equivale~e relation on the set of curves over

Jmor-

dst"

s

e

,ion 1-

is

be

F.

8

4.

Elliptic curves over 4.1.

Let

A

n:

and m

(Reference:

be an elliptic curve over

~.

gives a closed I-form on the Riemann surface

Weil [33J). Any differential

A(n:).

w

g

o 1 H (A,n )

(4.1.:'

w # 0 , its set of inte-

If

Given

gral periods W = {f w

(4.1.1)

Y

q.

(4.1. 6)

HI (A(n:),Z)}

£

i

j

y

forms a lattice in

l:c, and the map

A(n:) r--J n:/w

(4.1.2)

(mod W) (B,

If

is an analytic isomorphism.

(4. Ie ,')

Conversely, given any lattice

W in

n: , let The

(4.1.3)

g2(W)

=

60

L

-4 w

= 140 L

~g

are -:'10

wow

f

W#o g3(W)

I

l"

-6

homo_.1E

w

woW

W#o

4.

Jacc t'

These series are both convergent; define a complex curve

~

by the equation, q

= e 21

I

(4.1.4)

(4.: 1 This curve is elliptic, and Weierstrass I s parametrization gives an analytic isomorphism

E/W ~ ~(n:) z >--+- Ci'w(z),RT' (z» = (x,y) .

The holomorphic diff'erential

"' = dx y

pulls back to the differential

and W is its lattice of' integral periods.

dz

on

are

IQ

(4.

2

Thi

c

n:,

9

This establishes a bijection between {pairs(A,w)!~}

(4.1.5)

Given a lattice

W, the invariants of

(A,w)

c

6

= 216 g3(W)

11 = g2 (W)

- 27 g3 (W)

corresponding to the lattice

HO~(A,B) = {a €~.

(4.1. 7)

2

3

~

is another pair over

3

= c4!11

j

(B,v)

are given by

c4 = 12 g2(W)

(4.1.6)

If

W ~ ~}

+---+ {lattices

The degree of the isogeny corresponding to

a

V , one has

aW ~V} .

[V: aWl.

is the index

Two curves

are isomorphic iff their lattices (with respect to any choice of differentials) are

homothetic.

4.2.

It is often convenient to convert from the language of lattices to

Jacobi's q-parametrization. q=e 2wiT (4.2.1)

Iq I

Then

<1

If

T

)r-

E (q) = 1 - 504

6

L L

n~l

3

y

a (n)qn 5

3 =x

k

d

akin) =

Define a complex curve

This curve is elliptic. the pair

\

a (n)qn

n~l

2 (4.2.2)

is a complex number with

and the formal Eisenstein series:

E (q) = 1 + 240 4

are both convergent.

= x + iy

dtn

A

E~(q)

q

by the eqJJB.tion E6 (q)

-~x+86li""

(A • q

III

=

.~).

hiLs invariants

y > 0 , set

10

(4.2.3)

c4

= E4(q)

In boi

scalar.

c6 = -E 6 (q) t.>.

= '1. •

IT (1_qn)24

v

n>l If

(4.2.4)

A (~) , then '1.

w on

W is the lattice of integral periods of

(4.3.4)

W = 27Ti (Z fl ZT) .

Hence

The exponential map gives an analytic isomorphism

(4.2.5) z 1--+ e

4.3.

z

Tate has observed that Jacobi's parametrization gives a simple analytic

description of elliptic curves over lR .

Proposition with

'1.

4.3.1.

real and

!1:22!..

0 <

Any elliptic curve

I'll

is E-isomorphic to a unique

A/JR

"l! •

The period lattice

!mew) > 0

and

'1.

< 1 .

w

W of any real differential

real vector and is stable under complex conjugation.

n

A

2 Re(w) " :II: •

Hence

We may assume

on

A

contains a

W = n(Ztl Zw)

Re(w)

is either

with

o

or

12

Set

(4.3.2)

T

={

-~

w-l 2w-l and let

o < I'll

(4.3.3)

'1. = e < 1 .

21IiT

Since

Re(T)

The lattice of

dx 2y

w = {.2:i (Z '1.

= Re(w)

if

Re(w)

=

if

Re(w)

= 2"1

and

on the curve

fl :ll:w)

0

!meT) > 0 • A

q

if

'1.

is real and

is then Re(w) = 0

21Ii

(2w-l) (Z tl b )

I

11

In both cases, we see that

W

is homvthetic to

q

W via multiplication by a real

scalar. But for any two pairs

(A,w)

and

(B,,,)

oyer 1R

with period lattices

Wand

v (4.3.4) Hence

1 2

{" £

A

is JR-isomorphic to

A

q

1R: "W S;; v} .

The uniqueness of

g

may be checked similarly.

12

5.

The analytic theory of complex multiplication

(References,

5.2.

Lang [12], Shimura

[22]) .

actly

criminant 5.1.

Let

A be an elliptic curve over End~(A)

cation if the ring field

K.

[.

We say

is isomorphic to an order

In this case, the lattice

A

R

has complex multipli-

in an imaginary quadratic

W of periods of any

non- zero

differential

w is a projective R-module of rank 1. Assume that Then

End~(A)

0,

is isomorphic to

the full ring of integers of

K.

where

W has the form

modulus:

(5.1.1)

n

where

£ ~

and

~

is a fractional ideal of

determined by the image of morphism of j (A)

~, the curve

has at most In fact,

j(A)

h

~

K.

The isomorphism class of

in the ideal class-group of

K.

If

a

aA also has complex multiplication by O.

A

is any autoConsequently

Aut(~) , where

h

is the class number of

is an algebraic integer of degree

h

over

conjugates under

is the Hilbert class-field of

K.

is

~, and

H

The sec:

K.

= K(j(A»

If we identify the groups

Bc...l cl(K) -

Gal(H/K) points

via the Artin isomorphism, then the Galois group permutes the conjugates of

,]

j (A)

as follows:

Here we write

J ( !!. ) for the complex modular invariant of the curve 0:/$.

One

also has the formula for complex conjugation:

(5.1.4)

which gives .the full action of

Gal(RfIll) •

I

13

;ura

I J

5.2. actly

2

criminant

By (5.1.4), the modular invariant

.j(O)

is real.

In fact, there are ex-

real isomorphism classes with this modulus (by (3.3.1».

-D

of

When the dis-

K is odd, these curves correspond to the lattices

li-

ftic I

;ial

(5.2.1)

where

-I

V

=

is the inverse different of

K.

The first curve has real

modulus:

'4

I

= -e

-w/ID

is

jtoThe second has modulus

Both

curves have one component in their real locus.

points where

dy/dx = 0

On the latter, the four

are real, so the graph looks pinched.

A

'4

14

6. /Elliptic curves over p-adic fields

6.1.

F be a

Let

formizing parameter

F* so that

of

Let

~inite

n , and residue field

=1

v(n)

with ring of integers

k = R/nR.

A

A

over

k.

An equation (3.1.1) for

F.

a.

all lie in

1

in

R

(if

A

A

over

F is

A

A

of

F.

multiplication: for such curves

I

l

(mod n)

gives the equation

m

F

F.

•

A ns v(c4 )

If

v(t.) > 0

=0

the reduced curve

which is isomorphic either to

=0 • then

A

; in

In this case the non-singuOl

AC(F

A(

I.

fo

= c~/t.

j

A1 "

st,...u

a

v(j) < 0).

so

its modular invariant

v(j) ~ 0

will neces-

acquires good reduction over

This will alNs be the case when j (A)

v(t.)

is an algebraic integer.

A

When

has compleX v(J) ~ 0

we

have the inequality

(6.1.1)

(6. .

is minimal subject

is elliptic if and only if

(if

Ol

Conversely, if E

A

has bad reduction.

has good reduction over

a finite extension

0

The isomorphism class of this curve is independent of

form an algebraic group

sarily be integral.

on

(6.2.

A has good reduction over

v(c4) > 0) or to a form of If

v(~)

and

A

R.

The curve

has a singularity and we say

lar points of

v I

r, s, t

the minimal model chosen. this case we say

Normalize the valuation

!'

Reducing the coefficients of a minimal model for of a cubic curve

(6.

uni-

R ,

Such a model is unique up to a change of coordinates (3.1.2)

* Rand

in

Tate (28).

.

A be an elliptic curve over

to that condition.

u

~p'

extension of

called minimal if the coefficients

with

(Reference:

g: .•

the

it

o ~ v(t.)

< 12

+ 12v(2) +

~(~)

for the exponent of the disct'1minant of any minilbalmocie1. >. One Cao also associate to

A/F

the exponent

v(tl)

of the conductor.

This nClrl~ Fur

negative integer is ao isogeny invariant which measures the amount af wild ramification in the division fields of

over

F.

A.

! t i s zero if and only i f

A has good reduction

I

I ,

i

15

6.2.

We can define a filtration:

(6.2.1)

A(F) 2Ao(F)

2

A (F) '2.· .. 2 nAn(F) = (a) l n=O

v on the p-adic Lie group is

A{F)

as follows.

Let

-

(6.2.2)

PEA (k)} ns

ect and for

~

n

1

let

vex)

(6.2.3)

< -2n

v(y)

and

~ -3n}

tion

J

G of

I I

1 ~

1-

!

r r

where

x

and

~(F)

is the sUbgroup reducing to the identity in

AO{F)/~(F)

A{F)/AO{F)

yare the coordinates of a minimal model for

-

is always finite; when

v(j) ~ 0

over

F.

Then

-

A (k) , and the quotient ns

A (k). ns

is isomorphic to the finite group

A

Similarly, the quotient

it has order ~ 4.

In general, the

structure of this group is determined by the special fibre of Neron' s minimal model for :es-

A

R.

over

~ (F)

The sub-group minimal model for

is a profinite p-group.

A at the origin

0 A ' using

gives the addition law for a formal group re

the subgroups in the ideal

An{F)

,,~.

A

z

Expanding the addition law on a

= -x/y

of dimension

can be identified with the points of

as a local parameter, lover

.

R.

For

n > 1

A whose coordinates lie

We then have:

(6.2.4)

.ona-

ion

6.3.

Much of the local theory simplifies when

char(k) = p

is greater than 3 .

For example, we have the following result on local torsion. Lemma 6.3.1.

Assume

v( 6) = 0

1)

If

veAl > 0

then the group

2)

If

veAl = 9

and

and

v(j) ~ 0

A(F)/AO(F)

v(p) < tcP-l)

:!i.!!.!!!!.

is isomorphic to A(F)p = (0) •

A(F)12'

;:us

16

f!22!o A(F)/A O(F)

1)

Under these assum ptions ,

has order ~ 4 2)

A

AO(F)

is a 12-di visibl e group and

1.

i-ae

0

has a minim al model of the form: curve )\ (1.1.) )

AO(F)

as

= {p = (x,y)

A has a singu larity at Let

TT'

(i,y) =

vex) < 0

(0,0)

be a root of the equati on

u

and

v(y) ~ O} . actio

0

4TT

=0

and let

E

= F(TT')

Over

•

Th'

E

we can change coord inates :

I

(poJ

x = Y

=

X/TT·

6

y/TT,9 is a1

to obtain a model for

(

p I

A with good reduct ion:

geny cl'

10: ADy point of order

p

in

A(F)

~ore mapped to the subgro up

A (E)

lewton polygo n for

over

[p];'(z )

must lie in the sUbgro up

under the coord inate change (6.3.2 ).

3

E

lincet his polygo n begins at the point >ur hypoth esis that

AO(F ).

would have

(p-l)

integr al slopes

It is there-

i

(7.2. J

Hence the ~

-3 .

When

:i'

(l,vE( p)) = (l,4·v (p» , this contra dicts

yep) < t(p-l) •

onica~ "

polyno mj (Tate

:

isogen y

I

17

1.

.t-adic Galois representations 7.1.

Let

T.t(A)

action of

be a perfect field and let

F and l

curve over

Then

F

(Reference:

Serre [19J). G

= Gal(r/F)

If

A is an elliptic

is a rational prime not equal to char(F), let

is a free

E.t-module of rank 2 which admits a continuous

Z.t-linear

G

The natural map

is always an in.1ection. geny class of

A/F.

7.2.

F

When

Consequently the G-module

=~

A(~) ~ ~/W

and

V.t(A)

depends only on the iso-

, there is a natural isomorphism

ree

When

F = Il , the action of

plex conjugation on

When <

F

= kq

•

is a finite fieJ.d with

.

lit

The G-module

polynomial of

a.

T.t(A)

in (7.2.1) is given by the acHon of com.-

W.

onically isomorphic to a(k)

G on

q = p

r

G

is can-

a topological generator is given by the automorphism V (A)

t

is therefore determined by the characteristic

In this case the llla!> (7.1.2) is known to be an isomorphism

(Tate [30J); hence the characteristic polynomial of isogeny class of

elements, the group

A over

F.

a completely determines the

18

This important invariant may be calculated as follows..

endomorphism

of degree

TI

The curve

A

has an

q, which on the coordinates of a ·Weierstrass model is

given by

This is the Frobenius endomorphism; it is defined over

phism

~t

of

Tt(A)

t # p.

for all

Clearly

a

F

and induces a G-automor-

acts via

~t

on

Tt(A) ; some-

(

.

what deeper lie the formulae:

where

1T

+

v'

~t

=~

+

~

Det

~t

=~

0

~

-I

pla

= deg

is interpreted as an integer in

1T

teristic polynomial of The group A(F).

v

Tr

K(F)

a

1T

= 'l.

En~(A)

Conse'l.uently the charac-

is integral and independent of

t.

is precisely the kernel of the separable isogeny

(1-1T)

on whE

Conse'l.uently,

,1

(7.3.3)

Card(A(F»

= deg(l-1T) = (l-1T)(l-~) = 1

- Tr1T + 'l. .

by

,7,

One has the Archimedean ine'l.uality:

y

as well as the p-adic criterion:

Tr1T _ 0

7.4.

If

F

is a finite extension of

G = Gal(F/F)

and let

quotient

=Z .

G/I

~

(mod p)

a

A is supersingular.

~,

let

I

be the inertia subgroup of

be an arithmetic Frobenius in G which generates the

If t # p

then

a

acts on Tt(A)

I

and its characteristic

polynomial: has integral coefficients which are independent of factor: L(A/F,T)

= det(l-aT

t.

Define the local

pa

19

When A has good reduction over

n I

I

F,

I

acts trivially on Tt(A)

and

•

'S

L(A/F,T) = I - TrrrT where

TI

is the Frobenius endomorphism of

q = Card(k)

A

+

over the residue field

Card(A

places

v

2

k

and

In general,

,e-

7.5.

qT

ns

(k)).

If

F is a number field then A has good reduction at almost all finite

of

F

We may therefore define the integral ideals.

N(A) =

n Ev

V(N(Av ))

v

l.I(A) =

n

EvV(l.I(Av ))

and

v(N(A ))

v

where

.e,.

in F

,the completion of

v

by

is a prime at the place

F at

v

v.

v

and v(l.I(A)) v

are calculated

Similarly, we may define the global L-series

the Euler product: L(A/F,s) =

IT L(A/Fv,~-S)-l v

B,y

(7.3.4) this converges for Re(s)

>

~

The knowledge of

product is equivalent to the knowledge or Vt(A) isogeny invariants.

l

.J

as a

L(A/F ,s)

Gal(F/F)-module.

as an Euler Both are

s 20 8.

The arithmetic theory of complex mUltiplication

(References:

\

Serre-Tate [21J,

values

Shimura [22]).

8.1.

Let

F

be a number field, and let

A

be an elliptic curve over

complex multiplication by an order in the imaginary quadratic field

K

F

with

(8.:

3

Fix an

isomorphism:

(8.?4

(8.1.1)

By composing <6

with the action of

(8.1.2)

By (5.1) the field

t

Let

F

En~(A) on HO(A,n l ) , we obtain an embedding

(8.2.~

i:Kc--"F.

(8.2. (

must also contain the Hilbertclass-field

be a rational prime.

K II lilt - module of rank 1.

plex multiplications on

Via

6

the

Since elements of

H of

lilt - vector space G = Gal(r/F)

K.

Vt(A)

becomes a

commute with the com-

Vt(A) , the representation

centr: (8.1.3)

(8.

J.

is abelian.

8.2.

Let

v

be a finite place of

be the residue field of

F v

F

where

A has good reduction, and let

The reduction of endomorphisms gives an

inJ ection:

k

(8. j.

(8.2.1) an whose image contains the Frobenius endomorphism ment of

K

"v

Let

a

v

(8.3.

with

(8.2.2)

The map

1

be the unique ele.-

6 v (av )

v-a

v

="v

gives rise to an algebraic Hecke character of

F with

III

21

21],

values in

K.

More precisely, let

IF

be the group of ideles of

F

then

A

de-

termines a group homomorphism

with

(8.2.3) which is uniquely characterized by the following three conditions:

(8.2.4)

.ing

ker(x )

A

(8.2.5)

If

(8.2.6)

If

a

of

F

a = (ex)

= (a) v

is an open subgroup of

is a principal idE!le:

is an id~le with

a

v

=1

and at those finite places where

IF'

X (a) = lNF/K(ex) • A

at all infinite places

A has bad reduction:

s a

nThe Hecke character

X A

is an isogeny invariant.

centrated at those places where

!':A.

Its conductor

is con-

A has bad reduction; one has the formula

(8.2.7)

8.3.

t , put

For a rational prime

k

(8.3.1)

I.

•

•

and let lNt:F - - K t t

be the local norm.

Define the homomorphism

•

Xt,IF - - Kt

a 1---+ X (a) IllN (at)-l A t

where

at

is the component of

duces a contin\ll)us character image

•

K t

ot

a

in

•

Ft

Then

Xt

the idUe class-group

is totally disconnected,

is trivial on

Cp....r.,/F•

F

•

,so in-

Since the

Xt must be trivi$.l on the connected component

_ _ _ _ _ _ _ _ _ _ _ _ _ _111&.1_ _

I

_

t ~

22

C~ of the identity in CF . But the groups CF/C~ and Gal(F/F)ab are canonically isomorphic via the Artin isomorphism of class-field theory_

l

Therefore we obtain a

Galois character o

(8.3.3)

~

~ comparing the action of Frobenius elements at those places

v

i

l

where

-

A has f

good reduction, one obtains the identity

ple (8.3.4)

;, t

Thus the Hecke character

tations of Section

8.4.

X A

contains all the information in the l.-adic represen-

7.5. X.e. at the infinite place.

One can imitate the construction of

(8.4.1)

F

~

Let

= Fell

(9.

I

!l

( ,'.

~

j

,

;e

l

[.

phi 1

0

C"·...\e

* ---.. K* and let :INat :F,co (J)

be the local norm map.

Define the homomorphism.

. ,..ill

(

(8.4.2)

... ,

,

Again this is trivial on

F*

The composition * CF-K

(8.4.3)

XQl;l

gives two Hecke characters

X and A

duct of the Hecke L-series of

at

* * .... 11: xII: ,

X of type A

X and A

XA

A for O

F.

By

(8.3.4) the pro-

is the global L-series of the curve:

(8.4.4) . Hence

L(A/F.s)

satisfies a functional equation and extends to

on the entire cOlllplex plane.

a.n

e.nidytic :1'Unction

(9

Chapter 2

9.

Curves over

9.1.

Let

A classification

H.

H be the Hilbert class-field of plex multiplication by

K.

To each elliptic curve

the Hecke character

The former lies in a finite set

tion to

over

H*

A over

A over

Let

H with com-

0 we have associated two invariants: the modulus

phism class of

o.

K be an imaginary quadratic field with ring of integers

H.

j (A)

XA,I

H

E

*

-> K

H

J of cardinality h. and determines the isomor-

The latter is a continuous homomorphism whose restric-

is the norm; furthermore

X depends only on the isogeny class of A

A

H Theorem 9.1. 3.

1)

~

j

be an element of

continuous h"1!!9!!I!?!J'''ism whose restriction to elliptic curve

J

and let

H* is the norm.

X: I

H

*

--> K

l!lL!!.

Then there is an

A with complex multiplication bY 0 ~ H with

j (A) = j

~

XA,"X .{ '0','

p!ir

~l',ii¥ cM(,t~'

X A

determil1eS1;li&"i~ije#'c14slliot 'A'o~i k

(j{A)~l(~)d&terDdl1eBthe isomorphi~i;lail'ij'brA. ()vern.

isomorphic over 'H

iff they are iSOflenoullOv&t'

it'

W~.will prove ~l4s result in the next ,8~,t1on; varieties, s,ee, Del1gne [7] and Shimura

12~]~

'l'\iocurVes are

's.naisomOX1?llic over fOT

.atld, the

ii'.

generaJ.i,zations to abelian

24

such that e(,,)

for all

"g

0

'"

and

o

w

0

= "'"

in

1

R (A,U ) •

g

1/1 :

A are define d over

Since all endomorphisms of

W in

H, any class

may be repres ented by a contin uous homomorphism

1/I:G

The Artin homomorphism

.*

-4

ab G

h:IR/R -

0*

Sine

allows us to view

1/1

as a contin uous homo-

morphism

t=

(9.2.4 )

"'0 h:IR -K*

Sinc hi 'e

which is trivia l on the princi pal id?les . If

Lemma 9.2.5.

B

= A1/I

, ~ B has comple x multi plicat ion

by

0 E!l!!. H crv

.mX A· . )i·X B=

ChUo: s.D iLlQlllOrphiSlll

Proof. .Choose

over H: +:A ...:.' B Ol

CI8lU'ly B hu 1.nq..tipllcatiOIl bT 0 over cODlpCldtioll

NoY let

a:

pr11IIe

reduce s y;

,

0

,.1 . ~.s

all

endom orphism

I fOr'

1/oI\f, endC'!"~rP!llS1!!

; .., 01; A

Of~ ~'. ~ e~l'Jl~lli~~ &ctiOl1 CD

~ s~,.~tJ~;;~~~>9VerB. ! " ; ' £) ::.;+;.~ ,~\,:.;>;:~. ~', ,.,..:(."

BO(B,lh be

+'1. Q

ii

v

be e. phce ot

at:;~d'i:et"Q~(A)

t~ the

define

P'roben 1us

0y(B)

11 y

lIJ.:~'. ~"'tled' ~ . e." Unique·ei..itht~ l:it:'O'~iBii~(Af'1fIi16h' '.'

B Where' 'both

on

be the

A.

X A

,e.na.t

ti' '13' also 'l'he v&:z;ie •

~s·

8004 Teducti
ill e. s1m1l ar )ll&lIner~ ','l'hell the .ident ity'

(IM~;6r;' ;Ae " :,~:j,;;:~ji;fi;:t;k,~~~~~~, holds in

Encl.x (A).

It suffic es to check this (mod

morph isms is inject ive.

Ev) ,

as the reduc tion Of ende)..

There (9.2.6 ) reduce s to the identi ty

is a cJ

S~

25

in

Aut (A)

,

But if

where

q

is the cardinality of the residue field.

is an arithmetic Frobenius at

a

v

v

in

G, then by the definition of

Tji :

Since

av(k)

= kq

(mod~)

this gives

The action of both sides of

Since the characters

X B

and

(9.2.7) and hence (9.2.6).

(9.2.6) on HO(A,nl ) gives the identity:

!I!.'

X A

agree at all finite places where

A and

B

have good reduction, as well as on the subgroup of principal idE!les, they must be equal on all of

I

H

•

One has a similar result for "twists" by isogenies. curves over

H with complex multiplication by

If

A and

B

are any two

0, they become isogenous over ii.

Choosing an isogeny

.:A ---+ B

the 'assignment :'

-"t.

is Ii colltinuoUB l~cocycle whose co,hCllil101Losri 'cl,aSlI Clalls clul

be~ represellted

by a hOIllOllIOr:pM:.

AJmlln this

-----------_ _ -- -- -- -- -......

26

xB =,,·x A .L

so, b,y

holds in

* . Hom(I H,K)

Over

x, let

C

be a:rry ellipt ic curve

Let

j(C) = j .

H with

and

j

Given

We can now prove (9.1.3 )

y

* ---+ K t = x/x:I C H This charac ter is contin uous and trivia l on

H*

formu la

since

K*

is totall y discon nected

classe s, and comes from a it is also trivia l on the connec ted component of idele Hence t

Galois charac ter

redUt 1

ljI:G ---+ K*

equal.

Notice that the iIilage of

via the Artin isomor phism.

By (9.2.5 ) the curve

is compa ct.

and

j (A) = j (e) = j

mines

A

,:A --+ B mines

x = Xc • A

H.

up to isogen y over is ration al over

t

H

A

= Cljl

= X.

ljI

lies in

0* =

*

The identi ty (9.2.1 0) shows that X A

= XB

G

0 with

has comple x multip licatio n by

Indeed , if

as

Jl (K)

XA

9.

deter-

!l!. must be trivia l and

then

Simil arly, by (9.2.5 ) the pair (.l(A), xA)

deter-

A up to H-isom orphis m.

9.3.

Let

The grOllP

has mul'ti plicat ion by

Proof.

Let

v

0

*

Hom( IH,K) .

over H.

ot:

A

0 over .•

G~(HfIll)ac'ts~1~~~s~to·tc.ut\fei! If

C1'&

;~(ll!Il)

'that

H where A

II

Sine

t

modlll al

'tbe

Indeed con.l1l &ation by 'C1

be a finUe place .of

• ~ej:ll>et~e:ena_'J:p?1SDl

If

(9.4 .l

A be an ellipt ic curve wi'thc omple x mul'tip ;i.ica't ion by

H, as 1fQll as op the grOllP

for

gives 'an

has good red11c tion•

:1'~ue111&~~,:1l~'>~~~:!~ZlJf}

II

27

so, by the definition of

o(X ) A

By transport of structure,

B has good reduction at

o(v)

and one has the

formula:

= oexv Hence the characters reduction.

Xo(A)

and

X agree at all finite places where B

B has good

Since they are continuous and agree on principal idl!les, they must be

equal.

9.4. 0 =

for

Finally we need a description of the group En~(A).

Ho~(B,A)

By (9.2.5) this group is trivial unless

If we assume the Hecke characters

X and A

as a left module

X = X A B

X are equal, then (9.2.10) shows B

that

Since the group

,

Gal (H!K) .. cl(K)

operatessimp;J.yt%'aIlllitiveJ.ye>n tbe"et o f ]

of

mOdular lb.VlI:tiants, we ~ write

with

an tIltegralideal of

. LemIlla

2.4.2.

If

K

xA "xB and

apro,'ec~ive O-!!!Odule of rank 1.

J(B)" aa(J(A». then the ~O!1p ,H<>mu(B,A)

One haS an iSOmorphiSll!

.!lL

28 10. 'he re the deg ree of

.E!:22!.. ~.

$"

is equ al to

the gro up By (9.4 .1) we can com pute

a com plex emb edd ing By (5.1 .3) we may cho ose

s on Tak ing com plex dif fer ent ial

A

Des c

tl,,/IN!!:...

and

B

10 . .L

by ext end ing sca lar s to

H0"1!(B,A) "H e.- ,. 0:

by

suc h tha t

0

has deg re

wit h lat tic es

o

and

-1

&

res pec t i vel y ,

we hav e

gen y Sin ce the deg ree of the iso

<>

is the ind ex

res ult . [<>&-1: 0 ] • thi s giv es the

The au1 ~

'c

A

say

phi c to

T(j (A) ) =

in B

•

F

are is

c " over

F

29 10. to

Descended Curves. 10.1.

by

Fix an invariant

= ~(j)

0 and let F

has degree

h

of a curve

j

be the subfield of

A over

H with complex multiplication

H which it generates.

Ey

(5.1) F

and at least one real place; the field tower is therefore:

H

(10.1.1)

Gal(H/F) ,

ly,

Gal (H/K) , d(K)

F

The automorphism group of

H is a semi-direct product:

(10.1.2)

Gal(H/~) , d(K)

wIIere the c0lllp1ex conjugation

T

t.ott .A 'be an eUipticcurve "~\{j,~~_i.",_\,

sq A can be "descended to

#o,t1~:,ter

>4



acts on

cl(lC}

ove:rH

with C
by inversion.

O.

We

F;'hli:h :f~ is~r-

F" If

H. .

Theorem 10.1.3. T(3(A» ~ 3 (A)

The curve

AIl4. T(XA) • XA

/X{'J;>~'-')';'-'<:._<

.' frslot.

Ass\llllll

in F I since B·

A

-

CaD

>

if

anJ! _

B we also have

B are iSClllOrphic oVer H. j(A)

;J.t .

;;>,~'-

4(:8)

be descended.

T

co_s~. ass\lIIIll

F

A

.

xB • xT(B) • ThB) . by ·(9,3.1).

(j (A) .x.t) •

(HB)··'XB)~; i.

11es in F.

fiXed

by

Ues

Since A and

T

Then ve can certainly find a curve

C

30

~o e ~(GO'O*)

A = C~o with

H,

over

If we furthe r assume that

in the subgro up

*

1 HC(GO'O)

<"[>

T

(X ) A

.

= XA

Since

'!

~O

' then the homomorphism

= XA/X c

and

G,

in

c

lifts to a comple x conjug ation

lies

this gives a splitt ing,

ther Res

(10.1. 4) and the restri ction map F

Write

is sur,lec tive. is define d over

F

Then the twist

B

= c~

A.

descen de~

i.e., those

curves :

which acquir e comple x multi plicat ion by

0 over

H.

Let

(10.3

~(G,O*) which is repres ented by the quadr atic charac ter

be the class in

Osl(a/ F) ~ 0* •

e:G -

Theorem 10.2.1 .

0!!lY " ,.-

,->

Res(~) for some ~ e ~(G,O*)

and gives a descen ded form of

F

ellipt ic curves over

. and

=

We are now in a positi on to classi fy

10.2.

e

~O

it

1)

Two descen ded curves

A

and

are isogen ous over

B

F

.ll

XA ·, Xli

.,...... -'.-''-,', (·i·-;,:':_·.,'··... ~

j-1D.v ariAAt there are eJ52) Within tach F-isog enY class {A} with! ! f'iXed :._:-r/>:.-,-'-- ::"~--'-~'" -;~';~'/\ :~ --:?:f'::,;~:A· . -i~;': . . _ , ' _ ' , . ,:.:".\? ",.::~}:", 2:;i;;f "._ . " . "-.:,?:;; ;':-,;~'r:~~:-y:~';:~i:;,-:.:L A • ~ A curves repres ented bY. the c-'." . ,. ", . ,.,.... !Il chsse s. _. '_.. .. _ ' " , ',',".. actlY 2 F.i,!l!P,-.•"tPb!'l .." . " ," ", ,_. ".... . . .

:/;'1.-

ana

~.

curves becane isO!OOrphic over

a,

where the F-isogeDjY .:A -

-:.:-.'<.i-'

Proof'·

Choose aD

1)

iso~e\1f

If

XA • XB the curves

A

is

At>

_... ' '"

becom es a com-

/'_;"-

andll"~;' is6g~us';'er i{

.:1 -·B and let

•

•

(En'1l(A)8Q1) ... K

(9:1:3).

has

31

Conversely, if

2)

If

ther that Res

~

G

o

= 1

10.3. F

A and

= j(B)

j(A)

Bare isogenous over

then

A

= B~

F

~

for some

<

they are isogenous over

~(G,O*).

If we assume fur-

XA = XB ' then A and B become isomorphic over H (9.1.3). and

If

~

A

n

= <

with

H,

Hence

n = 0,1 .

is a descended curve,

,(XA)

=

XA

and the L-series of

A over

is given by

where

XA and

X are the complex valued characters defined in (8.4.3). A

This re-

sult is due to Deuring [8], as is the formula: (10.3.2)

By (10.3.2)

If has

v

A has bad reduction at all places

is a place of 2

F

where

A

distinct prime factors in

of

F

which ramify in

H.

has good reduction, then

A

is ordinary iff

q"

K.

v

--------------,---32 Th 11.

Ill-curv es.

A is H-isog enous to all of its conjug ates

"lIl-cur ve" if

aA • with a

A cannot be isomor phic to all of its conjug ates unless

Notice that

0 is a

H with comple x multip licatio n by

A over

We say a curve

11.1

g

1)

Aut (H) "j

Even

h = 1.

be to determ ine when the isogen y condi tion is quite strong ; our first task will IIl-curv es actua lly exist.

Since

A is a !Il-curv e if and only if

Lemma 11.1.1 .

a(x ) = X

A

A

for all

a

g

Aut (H) .

); it allows us to conThis result follow s immed iately from (8.3.1 ) and (9.3.1 chara cters. struct !Il-cur ves by constr ucting the corres pondin g Hecke

-D

Assume that

11.2.

= diaCK/111

is

Then the 'inclus ion

~

Combi ni Ql-cur

:I: Co.-+

0

in-

,.~

(11.2

duces a ring isomor phism.

'lJ./D7L = 0/.;::00 •

(11.2. 1)

.

K/III

The abelia n extens ion ~" (1l/D:&)

'

CQmposing

........ {±1}

..

- .. : '

,

:

. ., : ' . ; : -

'"

(11.2. 1) ~ives a ch!U'a cter with the isomor pbJ-sm ,;-,.;-,

H .not dividi ng

D with

, C1aSS-field~h~~ ,the ideal lRH/K.I!v .. (av )

satist ies a, : -', v which ".~":-

...

',i : .'" ..

:<

lIhen D '" 3 that

that

.-<..•..••

c __

"~'

some

-"0,

of vbe.... :>.a. ,;_....,~e Let:.>•.... /:·.t-:::·,;<' -...

<

11

corres ponds to a quadr atic Dirich let charac ter

this det:.ltl 'lldn"s,

by req¢. ring.

10-,

33 Theorem 11.2.4.

1) xD(a) xD(a)

There is a unique Hecke character

=l'IH/ K (ct) = IT

v¥ao,D

Since

if

ct

= (ct )

a

v(a ) v v

if

a

X : I - > K* "Which satisD H

is a principal idHe.

= (av )

is an idele with a

v

=1

for all vIOCl,D

a(xD) also satisfies the conditions of (11.2.4) we have

(11.2.5)

I

Combining this "With (11.1.1) "We see that Qj-.curves over

(11.2.6)

H

If

G = Gal(R/H)

X gives a canonical isogeny class of D

"We have a bijective correspondence:

ISOgeny classes Of} ... , _ _...., Jt-(G,O*)Gal(H/IIl) [ (Q-curves over H {A}

11.3.

When D is

~

the construction of Ill-curves is more delicate, and in

some cases it is impossible (Shimura [25]). that Ill-curves eJP.st whenever Pii!

D 18

3 (lIlOd'4). Wheu

andt,t1,

We leave it as an exercise to check either 'by

8

or by some prime

Chapter 3

Local arithmetic

Clear

In the remaining sections of these notes we will restrict our attention to elliptic curves

A

with complex multiplication by the integers

quadratic field

K

of prime discriminant -p.

theory, the class-number then

h

of

K

is odd.

Then

p

=3

1

0 of an imaginary

(mod 4) and by genera

We shall assume further that

p > 3 ;

0* = <±l> •

groups subgr

1

Galois

are a L 12.

A classification over

12.1. and let

F

F.

Fix an invariant

= lII(j)

j

unique £

J

of a curve with complex multiplication by

0

ment If

• Recall the field diagram: then

at the

(12.1.1)

12

F = 1II{j) gives

xp .

1

~

class.. . J

L _ 12.1.2.

Let

F8 '\

ll:p. K8 '\ ... V~)

~ ,\x

K;h-l) /2

.~

F8:lt"'m II

Then

a:(h-J,ll~"

r 35

(r-p) =

(p)

Clearly 'I

Ei

Let

G(Ei)

subgroup

I

for all

p

G(E ) i

groups

I

= K

H

=

E1 E2 • .. E" 2

2

2

E1 E2 "'Eh

i

be the decomposition group of the place

E

i

in

Ga1(H/~).

all have order 2 and form a complete set of conjugates.

Cl(K)

has

~

order, there are precisely

h

Since the

elements of order

Galois group, and they form a single conjugacy class.

Hence the sub-groups

are all distinct; renumbering them we may assume that

G(E ) = . 1

unique prime

E

which divides

p

and ramifies in

H

2

E = E 1

The

2

in the G(Ei )

Then

F

has a

A similar argu-

ment with the decomposition groups at infinity gives the lemma. If

then E

E

is the prime of

= discH/F'

at the place

12.2.

E

F

which corresponds to the unique embedding

Consequently, we see that .!!:!!Z descended curve has ~ reduction

(10.3.1).

Since the discriminant

-p

of

K is odd, the construction of (11.2)

gives us a distingUished isogeny class of lIl-curves over x

p

•

F <---+ ~p

F

with Hecke character

By (10.2.1) this isogeny class contains exactly two distinct F-isomorphism

el.asses. .;. '~-'-~

Theorem 12.2.1.

~

A(A(p)/F) = (_p3)

~(A(P)·/F) =e6(_p3) E!.22!.Let Then

There is a unique i-curye

(~)

The other curve' A(p)·

A(ll)

over

;;'";

F

with

xlj.(p) = xp

in this isoSem' SlaBs has

'

'

•

"'e

in

F.

36

is a ~-curve over

If

A

so

a(A/F)

F

with

X A

= Xp

then

N(A/F)

= ~.(p)

has support precisely at those places dividing

p .

Je

I claim that

(12.2.2)

for all

v

£.i

i.

Since

A

(a(A/H))

= vT

(a(A/H))

~i ' and by (6.1.1) it is enough to show that

clear, as ~ form of

(1

£.i

N(A/H) = (p)

has conductor

=6 it clearly has bad reduction at

v~ (a(A/H)) ,,0

A has good reduction at

~

exponent of the discriminant changes by a multiple of

(mod 6).

This is

(Serre-Tate [21]), and the

6 under quadratic twists.

By (12.2.2) we have

wr

(12.2.3)

good reduction at

V (lJ.(A/F)) q i

=6

vI:!. (a(A!F))

= 3,

lJ.

at

E.

e.

If

Ve(lJ.(A/F))

But the exponent of

1

cannot be

• the twist of

A

6, or 9 •

6, or else some twist of

=3

we let. A(p) = A .; if

Ve(a(A/F)

by theQ.uad.ratic.extension '•. H •

a "base point"

A would have

=9

37 Given this description of descended curves, :it :is"asy to check that the twist is a Ill-curve if and only i f

.. *2 (,.od ~ F ) .

6" 1

Hence we have bi-

jections: *

1"

IF*2 -

*U *2 III

IIll

-

{descended cUM-esl

U { III - curves l

To label Ill-curves we will use the set fields as coset representatives for curve

which becomes isomorphic to

write A ( p)

for

{dl

of discriminants of quadratic

* *2 III IIll A(p)

A(p)1 and * A(p) for

over

be the Ill1" ( /d)

A(p)-P •

Following (12.2.1) we

38

13.

A rational p-isogeny.

Cor

In the last section we were able to distinguish the two

geny class determined by .

X

by their local behavior at

p

E

~-curves

c

in the F-iso-

Here we shall show

how they differ globally and at the real completion of F • eou 13.1.

Over

H the curves F

A(p)

A(p) *

and

became isomorphic.

Over

(13.1.1)

HO~(A(p).A(p) * ) = {~

in the F-isogeny class of

X

p

we have

(13

Any

F-isogeny has degree divisible by p

endomorphisms Let

C(p)

±r-p

over

En~(p)

the two minimal isogenies

denote the kernel of the isogenies

= Hom(C(p).~) p

±w

become the

H.

is a finite group scheme of order

" C(p)

g

p

over

±W:A(p) --+

A(p) *

it : Then

F. its Cartier dual

C(p) (moe

is isomorphic to the kernel of the dual isogeny

oJ * --+ A(p) • ±W:A(p)

Theorem 13.1.2.

In Jihe category of finite /P'0up schemes over

isomorphisms:

w:

,~c

r--p t:....•::v

·c();)

Proof.

F we have

Since

becoaies· isomOrphic. to' ,:.:

(

Char(F). 0 •

:.

'l/p".. ov~

",~./,;'- ' > . i ; , ,--' ",

.',

\

phiSlll ,:Z/pZ _'·0 ·.over . F.~ ·we obibain squa (13.1.3)

a~

.-1

0

a(,)

~. we s 0"

'r

-

39 Conversely the Galois character of

C. When

th

P

E completely determines the F-isomorphism class C

c

=

~

p

,

the associated character

=

E:

gives the Galois action on

E:1J

P

-roots of unity in

This character is sur,j ecti ve, as

F.

equation is irreducible over

FE.

When

c= u

lIIk

cyclotomic ; hence we

we have

P

must show:

EC(p)

(lE::l) 4 = E

(13.1.4)

(.E!!) E -J c(p)

To show that

EC(p)

(or

Hom(G, (Z!p:ll:) * ) .

in

4

= E

is a power of the cyclotomic character

EC{P»

it suffices to check that it is trivial on the subgroup restricted to the larger subgroup (mod

f-P)

EC(p)

p

•

But when

is simply the reduction

of the p-adic character, p :Gal(F!H) p

which, in turn, is determined by ramified o1l.tside

AutOlll:ll: (T A(p» p

XA(p)

p

(8.3.4).

This shows that

p; to see that it il trivial onGal(F!r(~,p»

tdVialon all Frobenius elements rational prime 'i. ;; 1 (mod p).

But

Gal (F!H)

Gal(F!F(\1 »

e::,

in' this case,

A is a prime: of V

EC(p)

is un-

we must show it is R dbiding a

40

Since

C(p)

and

.-

C(p)

are Cartier dual:

.-

k + k

_ 1

'1'

(mod p-l) .

Since the two groups become isomorphic over

H :

s

Combining these identities t we see that either

k

or

if

¥

~ (mod p-l)

(mod p-l) ; the other is congruent to

To complete the proo~. we must distinguish the curves F •

is congruent to

A(p)

and

A(p) *

over

we see

or

. Note that exactly one o~ the groups

'I..

over

Hencl1t

and

is pointwise rational over the Bub-~ield M+ group o~ the real place o~ are

F.

M+-rational precisely when In

+

M

the prime

E.

p

(13.1.6) .•. - ,",",.'.'.,+ v q,(/l(A(p) . 1M .

»=

C(p)

(mod 8).

is totally ramified:

= ,

= F(~ )+ ~ixed by the decomposition

It s~~ices then to show that the points o~ p" 3

E.

=q

(J2::l) 2

We find:

r;ifP [: :: I

When

V

q

I

(6)" 9

r

ill poinh111e rat:10Iltll..

!;las ker unique real completion

ease

1\.

41

Theorem 13.2.1.

{

sign c6(A(p)) = (~) = p

Since the real embedding of

F

+1

if

p

7

(mod 8)

-1

if

p - 3

(mod 8)

-

corresponds to the homomorphism

L_-n,

1Il{J)

j I--->- j (0)

we see that. using the notation of (5.2).

over over

F:m.

FurtheI'l.ore. the curves

FE; they become isomorphic over

Hence the statement of Theorem 13.2.1' Theorem 1:>.2.2.

A(p)

2Y.!:!:

becomes isomorphic to either

A(p)

and

A(p)

'4

are not isomorphic

H. where the real plac,e of

F

is e'luivalent to the following.

Fit

A(p) =

*

t: ",

if

p

-

7

(mod 8)

if

p

-

3

(mod 8) •

;

..

A

is ramified.

42 14.

Local invariants and global torsion.

E

,

the 14.1.

~-

In the previous sections we have described the local behavior of the

F curve

A(p)

at all places of

is a finite place of over

F

v

F dividing

p

and

F which does not divide

~

On the other hand, if

p ,then

and the number of points on the reduced curve

A(p) A(p)

'r

has good reduction is determined by the

In par

Heeke character

call

It is now a simple matter to compute the local invariants of any descended

curve

B

.

v

= A(p)$

we have only to see how the invariants of A(p)

quadratic twists.

behave under

We summarize these results in the following table. Table 14.1.1

Local invariants of Kodaira S bol

B

theore

= A(p)$

T Comments:

III III*

B(Fv )/BO(Fv ) = Z/2 •

= (Z/2)2

10*

B(Fv)/BO(Fv )

1

B supersingul$r

0

I·0 '.

B ordinary _

ii

•

10 .

F

(&) = +1

P supersingul$r _ ..

B(Fv)/B~{Fv) .. B(?v)2

(~) P

Since '" -1

pro, C·

B

1

0

The sui

1 4•

vl2

3

6

{:8

dee,

I 8•

if.

II

if

+1 ber

locr'

a

f I

S'W,";-I

43 Similarly one has the following generalization of 13.1.2.

r-p

the kernel of the endomorphism

on

Let

C(p)W

denote

A(p)W , considered as a group scheme over

F. in

Theorem 14.1.2.

In particular, when A(p)W cally to

Hom (Gal (F/F), (Z!pZ) * ) •

is a Ill-curve, the group scheme

C(p)W

descends canoni-

Ill.

14.2.

As an application of our local results, we will prove the following

theorem on global torsion: Theorem 14.2.1.

1)

2)

Proof. Since

If

A is any descended curve

A(F) torsion --

A(H) torsion --

[

(Z/2Z)

if

2

splits in

(1)

if

2

is inert in

if

2

splits in

if

2

is inert in K

[ 0/20

= (Z/2Z)

2

(1)

First we will· show that the order of. A(F)torsion

A . bas bad reduction at

.e

K K

K

is at most

2 •

•

groUp' is&. direct

r:

.

of Type !Uor IU' the

pro~tj

I I

,,

The

.~b.-.sroup

, . . . ,; .

AO(F)

is a pro P-grOllP.

.e.

but ,l1O

. .'

p"torsion ,C41l exist

. deed. 117 CebOtar",'s densitT theorem we c~ find .

In ..

.t t

-1

(mod p)

where

A has good;

: -...

,..'

a prime.

..

-

a1oball7. In-

A. ot. F ",.th

jupersi~ar red~t1oil•. Sillce the num-

:.> --,-."

ber of points on the reduced curve

l~cal grO~A(F>.)

has no

A

is then t + 1

(whiCh is prime to

P-tor~io~. cons~4;u\ut~~ A(F)t~sion 1.1

p). the

isomorphic to

iii ~ll
auD-aro\lP ot

Z/ Z&.

SWJ.ar Brg\llllents show' t

'.

44

Over

When

p

=7

trivial. A over

H the 2-torsion gives rise to a Galois representation:

(mod 8)

the ideal

In this case, we have F

of .the form

splits in

(2)

= A(H)2

A(H)2

y2 = fix) ,then

fix)

0

and the group

(0/20) *

is

= ('11/2",,)2 ; i f we choose a model for splits completely over

H.

Hence vari

fix)

must have a root in F ,and

A(F)2 =

Z/zz. . d! e

Finally, we must show A(H) 2 ideal (2) is inert in 0

= (1)

and the group

equivalent to the statement that

£2

when

p - 3 (mod 8).

(0/20) * has order

is surjective.

A12.

3

Our claim is thus

(:

Since the question of surjec-

tivity is insensitive to quadratic twists, we may ass1lllle that at sOllle place

In this case the

A has good reduction

Then the reduction is supersingular, and A(k)

has odd order.

has

This gives a splitting

),

Our where ,

Al(R ) A

is isomorphic to the points of a formal group of height

integllrs ot'H).. ~.

,

This gl'Oup can contain no 2-torsion. ,,"S

2

over the

R.... 1 'e

1s

Le· o

•

T Chapter

15.

Restriction of Scalars

15.1.

Let

variety over dimension

~

A

4

Global arithmetic

(Reference:

be a ~-curve over

F

which is obtained from

h" [F:~]

if

G" Gal(H/K) B

=

Weil [32]).

TT

and let

B = ~/~(A)

be the abelian

A by restriction of scalars. we have an isomorphism over

Then

B has

H:

aA

l1eG Since the factors in the decomposition are mutually isogenous, the variety

B

has multiplication by a "matrix ring":

=TI

a a'

H""'l!( A,

A).

C1,0 t

Our first task will be to describe those elldomorphisms which are rational over Equivalently, we will describe endomorphisms of the functor on

R""""" B(a) .. A(a8~). Any a e

Let

R

be

UlU,\A/"L

K

K-algebras:

s ..

R8~

•. and

a

-----------------.-w-----r 46 runs through

G and

$

rWlS through

I

R=

(15.1. 5)

H0"1!(OA,A) H0"1!(oA,A) • 0 •

o£G Lemma 15.1.6. ~.

(15.1.2).

= Eny

R

The factorization (15.1.1) gives us a complete description of The elements of

endomorphism ring over

a

.

= «$0.0'»)

H.

En'1c(B)

En~(B)

are precisely the G-invariant "matrices" in the

But the element

into the endomorphism

p £ G

conjugates the endomorphism

= (W o • o • ».

pta)

The G-invariant "matrices" are thus determined by the

where

h

entries

$ 0, l ' which

gives the lemma.

15.2.

The ring

R

contains a subring isomorphic to

just the endomorphisms Lemma 15.2.1.

Let

?;root: • t.,

R

•

f;

t$

of

B with

/ ,

H-algebras

-'

S '"

HOIDn(aA,A)

and

•

m.~\I': But i f

be a basis of

Ho"1!(~'A,A)

•

"

P

'

III

EO

t.(S).o

<

EO

A(S)

a' a = aa' • we are reduced to a

Let

these are

0 = 1 .

.'

t.,(S) (p)

Since

En~(A)

is a commutative O-algebra;

t • • ~t sutfice~ to show that

0

0 =

=•

p~of'

. Then

0

a.,

:..•.,."..;,.",,.,,.

To show that

t•

-r

47

I

$* (w)

I

=

crw

h$

cr'

$' * (w) = h$'

with

h$,h$' < H*

Pulling back

aa'

w to

A

w

$

via the maps

0

a$'

and

$' o

a'

$,

we see it is enough to show that the identity

h

holds in H*

*

Since

H

H0"1!(

h

As an O-module,

th

•

h$' •

oa' A,A) K*

differ by an element of

so must differ bY an

$

~

II!

has rank lover

K, these two elements of

On the other hand, they have the same norm to

root of unity.

Since

R is projective of rank

h

h

K,

is odd, they must be equal.

(15.1.5).

Let

(15.2.4) Theorem 15.2.5. T

T

which ramifies over

E!:.92!:. R•

Let

is a CM-field of degree

h

~

K.

R is an order in

0 preciselY at those primes dividing h.

First we need the equations satisfied by certain special elements of

a-I

~

be an integral ideal of

K

.

and let

··a

~

±+: - A - - A be the two corre~

8Po~~ 1l!ogenies with kernel isomorphic toO/~ (9.4.;1). the oorrsspobdttlg K-'ena(ll!lorf)hism of

a~n

=1

t

(tl:l) .!.

= ~n

B.

then

t n ~

lies in

O.nd

.. t

Let

-e.

-

+~

be

-

TIle first ststeJllent follows from the defil1it1:on of checkll!d by extending scs.J.e.rs to BOw ohoose .(Oi)

".!:l.h

"

j

the second can be

a: ••

e. set ofindellendentgllnel.'"at§:t"s

1l.l,••••.!!r

·fotc.t(K) •

Let

'1'

is g"nerated over K by the elements t • and a "'i h . ; ... ) thsseelements satisfy the telations tAt'" Qt·' To see that '1''' K(t ......t j

then the algebra

t~

!!:.L

.!!r

48

Then

L(t

•.•. ,t

)

is an abelian extension of

~ this is just the Kummer extension of

!.l C£'(K) subgroup

K*(h) /K*h "--+- L* /L*h •

the ramification of the order

T

is the fraction field of

over

K and

T

Hence

T

L

L

with Galois group isomorphic to

which corresponds to the finite

must be a field of degree

R also follows from Kummer theory. En~(B)

is a ~-field.

Over

the ma.ximal totaJ.l.y real subfield

+

T

and

dim B = h , we see that

Ill, the variety of

T •

h

over

K;

Finally, since

B

is simple

B has multiplication by

1

1

I !

49 16.

The Ill-rank.

16.1.

Let

A be a descended curve over

finitely generated. and

~(A)

The groups

A(F)

Since the addition law on

A

is defined over

induces a Ill-linear automorphism of k € K

are

~(A)

V = A(H) 8

in L = K tl KT

Hence the algebra

As a representation of

0

F

K.

the generator

The algebra

L ,V

Endlll(K)

A illllo'iiOUffl!,"(Iif(jl;}\li,O.,(miOdli)

!:£eg!. ,By (15.2.5) the vector ll'lllLOe

A(iI) 8 III • B(ll) 8 III ~"N.:<":~~L~'f;"

over

is a module

i'

over Ill.

(16.1.1)

2h

and

= M(2,1Il)

K.

(inod2b,) , •

CM-tie1d T , of degree

K also

decomposes as a direct sum of simple modules, all

When A is a IIl-curve we have a much stronger result on the railk.

16.1.3. .!!

of

En~(V) .

is isomorphic to the matrix ring

isomorphic to the minimal left module

Theorem

T

we find:

(16.1.2)

(16.1.4)

A(H)

In (14.2.1) we determined their torsion subgroups; let

acts Ill-linearly, if

Us:4!g

and

denote their ranks.

Proof.

Gal (H/F)

F.

III

and

Let

B=

'1l(A);; 0

fF/u!.

50 F

A be a Ill-curv e over

Let

16.2.

B = R / (A). F IIl

and

The finite ly gener-

ated abelia n group A(H) = B(K)

R=

is a module for the algebr a struct ure of

K as a module over

A(H) 1II

0

A(H)

determ ine the struct ure of

In the previo us sectio n we studie d the

En~(B).

R 1II K = T.

0

In this sectio n we will

0

at certai n finite places of

pol G

Let

= Gal (H/K)

TheorE llll16.2 .2.

O/"O[G]"

algebr a

1)

,,=.r.:p.

The algebr a

P

1)

R/"R

is isOmo rphic to the group

,8

is I

!~

Let

P

is isomo rphic to

n(A)

copies of

R/"R.

be a non-t rivial elemen t of the group

is define d over a quadr atic extens ion

. L = H(\lp )(Id) .

= R 1II0 0/,,0

A(H)/" A(H) = A(H) 1II0 0/,,0

the regula r repres entati on of

!3:22!.

(

~pZ[G] •

The module

2)

and

L

of

H( \l ) : p

if

A

A(H)"

= A(p)d

then

A(I

Using the decom positio n II

(11

with

0.. ~p'&. S:lnce ~(aA.A)

.1

a... 0/..

p • the

map • l--- a. induce s an 18OlllOrph1S11l:

(16.2. 4)

One check sthllt this lscllllO rplnsm

s

51

=~

R/wR

B~(aA.A)/WBO~(aA,A) • a " , ' ~ O/wO • a • a

a

2)

in

Let

R reduc es to

R/wR = O/wO[O].

in

a

satisfies the polynomial (mod w) t

~

= 1 ; the element t a = $a 0 a a If a -has order g in O. then t

be an isogeny with

(xg-o)

in

o~ ~

~

EndO(A(H») ,where

(0) = ~g

and

0" 1

Since this polynomial is irreducible t the characteristic polynomial of

on A(B) 8

0

K is

Then

polynomial on the lattice

A(H) So Op

~

has the same characteristic

in A(H) So K ; reducing (mod w) gives the p A(B)/wA(B): (xg-l) (h/g) 'n(A). Since this

a on

characteristic polynomial for

t

n(A)th power of the characteristic polynomial of a in the regular repren(A) sentation V of O. we must have A(H)/wA(H) = lEI V (Serre [20]). i= 1 is the

Theorem 16.2.5.

Assume (2) splits in

1)

The algebra

R/2R = R So 0/20

2)

The module

A(B)2

= 0/20

!i:22!. generates

Let

aA(a F ) 2

is isomorphic to the grOUP algebra

A(B)/2A(B) = A(B) So 0/20

~ n(A)

1)

K.

P

is isomorphic to the direct sum of

copies of the regular representation of be a generator of A(F)2 $;A(B)2

and for any

in. Boma'(a;A,A)

•

0/20[0].

we

~

R/2R.

(14.2.1).

Then

ap

.write

(16.2.6)

vith .~. e 0;20

Since

Homa«JA,A) ,.-"

.

contains.isogenies of odd degree, the map -".-

',.

-,.'C,;

",

........ - -"",',

.....

,

a.

-',''- \ .'. '-:i:-;:{.c'r: -.,/-)i,r;~:~·s; ,;,~: .""<'"

• j '..

R/2R -:-' 0/20[0]

,

52

2)

We will show thst as a module for

Since

G stabil izes

semi-s imple.

But

trivia lly on

A(H)2 •

Let

then

A(H)/2 A(H) Let

A(H) t

= $

h = Card(G )

= A(H)2 $

N/2N , where

(J

reduce s to

(J

in

R/2R = O/20[G]

(a) =!!. and

a" 1

Since

a" 1

A(F)2 ; hence

in

If

(J

hss order

is

G acts ex

Hence the chara cteris tic polyno mial of

g

in

t

(J

on

N/2N

is

Tra" 0

(xg-l) (h/g)2 n(A).

2'n(A )th power of the chara cteris tic polyno mial of 2'n(A) V. fD = N/2N have we G, of V on regula r repres entati

(J

A(H)/1IA(H)

module s for till! 1'ull group

Aut (~)

and

A(H)/2 A(H)

(when

2

splits in

p

v)

on the

i=l

The spaces

l

Since this polyno mial

(mod 2)

Since this is the

~.

G,

and satisf ies the polyno mial

in End-(N ) is "7.0 !!. (mod 2), Na" 1 (mod 2) and

is irredu cible, the chara cteris tic polyno mial of g 2g (x _Trax +Naf.! '/g)n(A ).

A(H)/2 A(H)

N is an R-stab le free subgro up of

!!.

!!:. !!:. (xg-a) in EndO(N) , where

(mod 2).

P

and fixes

A(H)2

$ :(JA ->- A be any isogen y of odd degree . 0

G on

is odd, the action of

II

is the direct

copies of the regula r

2' neAl

sum of 2 copies of the trivia l repres entati on with repres entati on.

A(H)/2 A(H)

:r,!2Z[G] ,

'1

K)

are

In the first case

HE

(l

0'

", t

(

.

53

17.

The first descent. 17.1.

Let

A be an elliptic curve over An = ker

H-endomorphism of

A , and let

exact sequence of

Gal (L/L)-modules

o --+

H, let If

1T

L

rr

be a non-trivial

is a field containing

A (L) --+ A(L) - - + A(L) -

rr

H , the

0

rr

gives rise to the short exact sequence in cohomology:

~(Gal(L/L),Arr (L) -

0 - A(L)/rrA(L) -

Applying this to the situation where

L

=H

or

~(Gal(L/L) ,A(L») rr

L

= Hv

--> 0

(the completion of

H at

v), we obtain the following commutative diagram with exact rows

A(H)/rrA(H)

0--+

1

(17.1.1)

o -Th(H v

Here G· Gal(ii/H)

v

and

~(G,Arr)

~

~(G,A)rr

-

!Res

)/rrA(H ) v

G

v

n~

Vv

iRes

'lJ~(Gv,Arr) -l!~(Gv,A)rr

is the decomposition group

Define the 'Se1Jller group for

".' S,,(A/H)

to be

t1n#e'~1lp &lid the resUlting injection

can be

u.a84 tc

A(H)/"A(H)

bound t~e rank of A

a

0

Gal (Rv/H ) v

n~

Then S,,(A/H)

v"

is a

S,,(A/H) . ,,'

The calcu.1.ation of S" (A/B)

"mit. iiifth~ first' " ....descent. " . Define the Tate-Shafarevitch ll1'Oup W(A/H) t"1on map

I

-

t~esu.1lgroup of al(G,A,,)

'l/hose elements, under restriction, lie. in thei.mage 01'

o-

-0

-'.-

to

. '

"'--'

'.

is called

'-

be theternel of the restric-

54

Then a simple diagram chase in (17.1.1) shows that

coker a ~

U](A/a) n . Here is

the full picture:

A(p)

Gal I tha+

o

sem_ .

1 W(A/H)

n

->-0

the: .;

1

WhE

e

a(p)

17.2.

We will study the first descent only in the special case when

is a Ill-curve and scheme

A"

A"

K so that

to

,,=,r.:p

n

can be defined over

A"

TIle group ·Ga.l.(ll!K) Ga.l.(H/K)

,or

= "p~

=2

Ill.

and

p

=7

(14.1.2)

By

K( Icl)

twisted by

(mod 8). and

if

2

= A(p)d

In this case the group

(14.2.1)

"

A

we may descend

=-p

1lhen acts naturally on

is prime- to.....

2p, the restriction map in,<1uc,es·.~ 1!!,'~.rphi'lm

..

-.(

(17.2.2)

In Section 16;2 Theorem 17.2.3.

The map

A(H)/"A(H)

).

J

r(Q.A~) "iB:~"~~rW!ism

of SUbgl

Gal (H/K)-moclules • (1 ' .

.,·~i~;:;;~~t, be a point in A(H) class of

P

(mod "A(H»

P"= and let

11'-

A(P)

R denote its image in

r(G,A,,)

Then

55

A(P)

g ~ g(R) - R.

is represented by the l-cocycle

H~ = H( VA(H)).

Gal (H/H) ,where H~

that

is a normal extension of

semi-direct product:

B.r

Arguing as in the proof of

Gal(H /K) , Gal(H/K) ~ Gal(H /H) .

w

w

the cohomology class

cr(A(P))

where

Gal (Hw/K)

in

(16.2.2), one shows

K whose Galois group splits canonically as a

our choice of descent, any

cr = cr x 1

This map factors through

cr

in

Gal (H/K)

,:crA --+ A

cr(p) =crw(cr(R))

in

Aw(K) , and

is represented by the cocycle:

and

g

Gal(Hw/H)

€

On the other hand, the residue class ,(o(p)) ,where

acts trivially on

o(p)

is an isogeny with

is represented by the point

B,

=1

(mod w)

(16.2.3).

Since

crA(H ) , we have

w

,(cr(p)) = ,

Hence a cocycle representing

0

A(cr(P))

°w(cr(R)) = w

,(cr(R)) •

0

is given by:

g f-->. g(,(cr(R))) - Ha(R))

II ,(ga(R) - a(R»

as

,€

En<1i(A)

a8

•

a =

I ,

0

a(a-lga(R) - R)

U

.

a-lga(R) _ R

17.3. subgroup of

If

,,=.;::p

Sll(A/H)

or

". 2

on. which

andp

Gal(H/K)

=

0

(mod 8)

acts tr:l,vially.

B,

let

acts trivially on· A",

Sw(A)

denote the

B.r (17.2.2)

in1"",tion on

56

the subgroup of invariants:

a.. (A(H)/uA(H) )Gal(H/K) "--+-

Su (A) 1 '

The calculation of

S (A) u

and W(A)

u

= coker

a

is much easier than the calcula-

tion of the full Selmer or Tate-Shafarevitch group, tion for the ~-curve A = A(p)

in Section 22;

gi".

We will perform this calcula-

for the general case see [lol,

By

(16,2,2) and (16,2,5) this calculation is sufficient to bound the rank; indeed:

(17.3.3) r

ank

0/,,0

(A(H)/ A(H»Gal(H/L) =

"

[ 0'" n(A) + 1

2

= -p

if

u

if

u = 2 p "

7 (mod 8) • (li

Wh

81 ( II

1

If

57 18.

A factorization of the L-series.

18.1.

Let

= RF/Ill(A)

B

A be a Ill-curve over

is defined over

F

By Section 15 the abelian variety

and has complex multiplication by

III

T

over

K

It

gives rise to a Heeke character (18.1.1)

which is Let



= Gal(T/T+)

equivariant (Serre-Tate [21])

.!. be a prime ideal of

and only if is the case,

The character

K.

is unramified at

is unramified at all of the prime factors of .!.

X

A

XB(!)

is the unique element of

*

T

in

~

if

If this

H.

which satisfies:

(18.1.2)

When restricted to principal id?>les,

leB

induces the natural inclusion

*

*

K <--,. T

Define the complex character:

X~ * I K - - (Tea:)

a

t--+

xB(a)

-1

a~

•

,~iiice this boaolaorph18111 is trivial on K ", • itgi...e8 rise to (i) . '

2h' Heeke characters

.. XJb"j;~! ~~;,\~,:",,:

,- ," \i.//'

";<~;_;'.f

',_ ,_,?"._~,"<':. "'-',T::

(18.;L~:U·, ." ' ' ......., ..

-

....

-::.

,-

,

.',

-',

.-

LiLbe1.th~C
izl&liuivia~t reel

'or' XBt,i'l.1'l~"~

lllllbed,Ungit ot '1'+

that" .

Pi

,ot

,.'1'(8~ t1i&t

and~2h-.i ~

Pi

{Pp •••• 0 T.

P~}

induce the

'l'he -equivariance

58

Furthermore, the L-functions of

B over

~

and

K are given by the formulas

(Serre-Tate [21], Shimura [25]):

jU ,

L(B!~,s) =

(18.1. 5)

Tr

k =

L(X (i) ,s)

i=l

(0)

2h

(18.1.6)

L(B!K,s)

= IT L(XB1

B

,s)

of

= L(B!~,s)

2

co

.

i=l

vani

Since

L(RF!~(A)!~,s)" L(A!F,s)

, we have obtained a factorization for the L-series

of a IIl-curve:

!!!!.t

Theorem 18.1.7.

A be a IIl-curve over

F ~ B" RF!~(A)

Then

L(A!F,s) .. L(B!~,s)

where

X is a Hecke character of B

K with v'9-ues in

T .

Finally, we note that the quotient characters: (~8.1.8)

are precisely the

18.2.

Let

h

ideal class characters

A be a III-curve over

F

ot lC.

and

B"

RF!~(A). ':\'he toUoirlnB ~onj~,"' h

c',

ture ralates the factorization

(18.1;7)

11' 'L{xii ) .s)

L(XA.a-}\"

to the falitorlza-

·i-l tion

(16.1.4):

n,,(A) .. b • n(A) • -'"

•

Co!!.lecture 18.2.1.

-.-)

The order of

L(x~i).s) !1.

soo 1

is>indep~cl...ntot·i

wh

59

The first statement has been generalized by Deligne to a conj ecture on the con-

jugate L-series of a motive over k = K and of

i

E = T

By

(19.1.1)

f~i)(z)

the parity of

(20.1.3).

vanish simultaneously, if at all.

,,

with coefficients in

We shall show that the L-series

conjugate new-forms

whenever

k

neAl > 0 .

L(X~i) ,s)

E

[7].

In our case,

ords=lL(X~i),s)

is independent

are the Mellin transforms of

By Shimura [26], the

h

values

Furthermore, Arthaud [1] has shown

L(X~i) ,1)

L(X~i) ,1)

= 0

60

19.

The sign in the functional equation 19.1.

Let

Tr

(Reference:

A be a Ill-curve over F.

Tate [31]).

By (18.1.7) we have the factorization

L(X(i) ,s) where X is a Hecke character of K. In the next two B i=l B sections we will study the analytic behavior of the h conjugate L-series

L(A!F,s) =

Since

X depends only on the F-isogeny class of B without loss of generality, that A = A(p)d with (p,d) = 1 .

Theorem 19.1.1. A(i)(s) =

(19.2.

(19.2.

A , we may assume,

Ind'

.!!. A = A(p)d ti1h (p,d) = 1 and B = RF/uf ' the function

i

plies

Fine 1

(Pd/2w)Sr(s)L(x~i) ,s) satisfies the functional equation:

with • sign d • I

f!22!.

Since

Ix = (;:P'd), we have

Furthermore the character Hence the terms

s

i

X~i)

has type

2 and (Pd)s = Ms /

(1,0)

(2w)-sr(s)

when restricted to

* K..

= <±l>

in 17.1.1 are precisely the ex-

ponential and infinite factors in Heeke's functional equation for

L(X~i),s)

To

cOOJe

corres

complete the proof of this theorem we must Bhov that the global root number sign d.

We

! c = ,

(19 !,

Ie" Ill-c'''''

61

w

=

wv = 1

if

00

v

%pd

(19.2.4)

if

Indeed, we may choose an isomorphism

plies (19.2.2).

cOv

with

= (f-v )

so that

(19.2.3) is clear as both

Finally, in (19.2.4)

(f) -v

Xv

and

is the conductor of

v ,and ljIv

• V

vlpd.

K v

Xv'

are unramified at c

is the "canonical additive character" of

is at most quadratic when restricted to

= <±1>

the character

character of

If

vlt"

p

Xv

v

The character

it takes values in

* p(T)

1 be the quadratic character OfJII~

ljI

let

2 • 1) w = (-) P P

ov :

K

=

always restricts to the unique quadratic

r.

corresponds to the abelian extension

Theorem 19.2.5.

.*

0 v

is an element of

If remains to compute the product of the local root numbers.

Xv

v

IIlt (,d).

i

Then

Xv

= ljItd

0

which

*

lNIC.v/lllt on 0v

•

--.,.'- .. '

"....

Vp = (,r.:p).

;;

We may take

62

and

= x(p) = x

1

~

(php (p)

= ill p Xp (p) = Xp (p)

making the change of variable

b = 2a

wp

=3

Since

p

gives

1) •

= Trw

.

If

t ; p. tid

but

v

vlt

We ~ take

c = t

gives:

=

4), the Gauss sum in the numerator is equal to ilP.

(mod

Now assume

Subst.ituting into (19.2.6) and

%d

If

t

t;

2 ,then

then

Xv

$~

is trivial on

~t*

is tamely ramified and

and

This

$~(-1)

$~(-1)

=1 =

= Ce1 )

.

in (19.2.4); I claim that

(19.2.7)

Again the product on the left side is independent of the Ill-curve chosen, so we aSSume that

t

A

= A(p).

= (-) IT Xv(t). p

Then

1

= X(t) = x~(t)~(t)1f vlt

Xv(t)

~

= JfL(~):rr Xv(t) p

=

vlt

There are two cases to consider:

vlt

a)

•

63

b)

(~) p

= -1

so

(.t)

is prime in

0

I

w.t

aEJFi'2 .t

= X.t(.t)

(II,,) e2~iTra/.t .t .t

.t-l

-( I

= (-1) a=l

where the first identity follows from the theorem of Hasse-Davenport and the second follows from Gauss' s determination of the sign. Hence, for all odd d

l

we have shown

When

~/l+8~ to consider.

there are three characters of

.t = 2

divides

This can be done by hand,

and we leave the proof to the reader.

Corollary 19.2.8.

Proof.

w = (g) p • sign d .

By (19.2.1)

w

=1T w

V = wm •

V

-1

w= 1

I 2 =H p

•

W"p

(TT wv ) vi! .

•

SUbstituting in the

1

64

20.

!i-curves and modular forms 20.1.

For

N > 1

(References:

Shimur,. 123J, !24]).

let c _ 0

As a discrete SUbgroup of

PSL OR) 2

the group

(mod N)} •

ro(N)/<±l>

acts on the upper half-

plane, and the quotient may be compectified into the complex points of a projective curve

XO(N).

The curve

field

~(j(z),j(Nz»

Theorem 20.1.1. defined over

E!:22!.

XO(N)

J

O

~

with function

(Shimura [22J).

There is a non-trivial rational map

F.

Since the curves

A(p)d

without loss of generality, that Let

has a canonical model over

= J~(p2d2)

and

A(p)-pd

(p,d) = 1 .

be the Jacobian of

Let

are F:"isogenous, we ~ assume, A = A(p)

o = XO(p2d 2).

X

P t--+ [pHi-]

over

and

B = ~/Ill(A) •

Since we have canonical

maps:

Xa'"- J O

d

III

65

tyists all satisfy functional equations of the appropriate type (19.1.1), it follows

f~i)(z)

from a result of Weil [34] that The character of

rO/r

l

is a ney form 0:1' weight

is trivial in this case, and

f~i)(z)

2

for

rl ( p

2

i).

has a Fourier ex-

pansion:

(20.1.4)

where

q = e

2triz

~himura [23]).

By a theorem of Shimura [24], the abelian variety

canonical factor

Since by

B and

Toyer

B of dimension O

over

III

"0

has a

with

III ~ have complex multiplication

K, they are jQ-isogel\oUB (Shimura [25]).

Qu.estion 20.1.5:

This isogeny exhibits

as

B

"0(p2d2) •

Assume

The ne;, forms

wh1cb. ~e defiDed Over the .:".",

h

B have the same L-series C1'ler O

a rational quotient of

20.2.

generate the field

The Fourier coefficients of

d

=1

Is the abelian variety

B

= IT/rI(p)

f~i)(z) co;r~sPondto~l"'rphiC ditfer8nti~s 011 !tq normal closurli 01' iot: in 1ft and are etBenfOl:omS .for the

";.-~.<:'

action of .~(Bo) 8 III .. T.

If we let

."::::'O:-.;.',::":·,::;·::':::::-':-·.>:.i\'::-;:::.'-."",:',:;:,

.

".

.' 0

3.

then the corresponding differential is defiDed' C1'ler III 8I;ld senll1'ates It (BO,1l ) as a "module over the Heelte algebra

En~(BO} 8

of 9 (Z) are all rational integers. B

In tact:

... T+ • The Fourier coefficients

66

(r

(20.2.2)

!!.-l ]N.!.=n as the trace of

20.3.

XB{!!.)

Assume

is zero unless

d = 1.

!!.

X {!!.» B

qn

is principal in

Gt{K) .

We can normalize the map

t Ove:

up to sign by taking it to be a covering of minimal degree over "O{[i~])

= 0A{p)'

I f we assume further that

functional equation of

A{p)

is

-1

(mod 8) , then the sign in the

1''' 3

and the map

F with

"0

factors through the projecw~-tl

tion onto

(Ligozat [13], Mazur [14]):

i

2

a

Xo~~ X

sp

l"t{p) ~

11'

. spUt

OJ V I

A{p)

I' " 3 (mod 8) •

" of .

Ligozat has observed that JO(p) x Jnon-split (1'), quotient of

JO(p)

Xnon...split <:1')

(lC/S [1) ,Iter

J split (I') Since

A(p)

is isogenous to the product has potentially good reduction, it is not.. a

and must therefore be a quotient of

Sinpe

and t

,!

~~. " non-t.rivisJ. .rationsJ. ;pobt

,l't .~ -",.".,;.".

()~n a f'l!J'ther pax·""".trize,ti.~ti'ClVil,l'

I

1

1

e

~(

·i.,

Chapter 5

The ~-curve A(p) •

In this chapter we will study the curve this ~-curve is defined over Over

21.

H

= F( 0)

Let

F ; let

LI(w)

= Z [l+~

= a v (w)

W

A W

v

andlet

v

which extends

is a holomorphic I-f'orm on the complex curve

has type (1,1)

of the torus Let

be the discriminant of

£:

w

in

F

]

a. A

v

= A(p)

* v

a: K '-+ IC • and for each complex place

be the unique complex embedding at

w v

til

0

(12.2.1).

p

w be a non-zero differential of the first kind on A

Fix a complex embedding

v

Recall that

and has good reduction outside

it acquires complex multiplication by

which is defined over

v

= ~(J)

in more detail.

Periods.

21.1.

a

F

A(p)

of

H let

The differential

= a v (A)

•

The form

we will compute its integral over the fundamental class

A "(H ) •

v

v

(Zip) * -

r(z)" (

o

t

<±l> Z

e-t dt t

be the quadratic Dirichlet character be Euler's

r-function.

The following result is due

by D~a.:l.gne.

91lQWJ,a. &lldSelberg, and was shown t,o \lie

O
identity

IT F(~" cl(x)

, (el').L""

IT f(c!p)12£(C)

O.c.ti.on on lattices in

IC.

Let

68

Lattice (w

(21.1.3)

v

=n

)

~

with

n

a

in

~

~

Then

(21.1

a: •

From (21.1.3) we find after

(21.1

Taking generators for

Cl(K)

of the form

-1

-1

-1 }

{O'~l'~l ,~,~ .... '~-1'~-1 2

IT IfA (H )wv v I~

(21.1. 4)

v

Since

fJ.

h

W

v

is a modular form of weight

v

I = TT

~

~

12, (21.1.3) also gives the identity:

w ) v ~

12 = fJ.(a)n- a

-

Now

CODap~1e

NB/IllfJ.(.. ) • F(!.) .• .. cl(lC),

(IT. ci(K) '.',' ',.,

I

to .....lE

Taking the product over our special generators gives:

IT

2

n i l . (,Ip)h •

Cl(K)

= fJ.(Lattice

(21.1. 5)

we find:

Cl.

il )-12

~ !.

fact t

\ .......~ ......;"

L

quoe

roots

Note, Define

The curves

A.· A(p)·

and

h!. whel

•

69

Then

=a

(h) .!

and

(h )

t

~

= h T,!. ( )

One has

(21.1.7)

(l

~,as

after extending scalars to

(21.1.8)

well as the cocycle relation

''b

h

~'1?

=h-

a

h-

-1c

We can obtain a sharper result on periods by using our knowledge ot

21.2. A(p)

a

at the real place ot

F.

~ m

= I

Lemma 21.2.1.

(c/p)

O
~ m is integral and m =

¥ _~

&(c )=1

f!:22!..

~

Dirichlet's class number tormula

-h =

I

dcHc/p).

Adding this

O
~l

to the identity

= I O
Z[1/2]

and

fact that

h

(c/p) , we obtain

m

-K* •..•C'

11" I vi" y

III

,....

v

v

b

Since

11-8

Theorea. 21..2.2.

-'.

2 = m.

is odd ).

q1lO~i~t'iiesin

where

4 -

Z[l/p] , it is actually an integer (this gives another proot ot the

Let us write

ProOf.

~h

~

IT

r(o/p) I (2lrd' where the intesral O
the definition ofA(p)

is a tractional ideal of

F.

(12. then'

70

If we put this in (19.1.1) and use Euler's relation

r(z)r(l-z); ~/5in ~z

2.

and that

(19.1.5) we obtain: Swinne:

• p 3h/2b 2 = (

(21.2.3)

IT r (c/p) / O
dc )=1 If

_a

+1

Furthermore, (mod 8)

we may choose the sign of

III

is real when

(13.2.2).

p" 7

nand a

(mod 8)

na -1

so that

n.!!. = N!!. • n~-1

and purely imaginary when

p" 3

(21.

Taking the product on the left hand side of (21.2.3) over our

special generators gives:

TT

In

Ct(K)

pa~t

Hence

b2 • (

llr n

)2 ct(K) !!.

= (lli

r(c/p) / (2wi)m)2 (j:p)1-3h •

O
Taking square roots shows that

IT

TTo ct(K) !!.

r(e/pl / (2wil O<e

" an::r

But

•

K

p:r;'~uet

of integrals

IT f "' vl~ y v

differs fran

m

.

ITo

ct(Kl·!!.

v

•

For a discussion plication by

of

the periods on an arbitrlt.l'Y e+Liptie curve

0 see

[9J·

..

71

21. 3.

Assume

that the groups

A = A(p)

A(F)

and

with A(H)

p" 7

(mod 8)

are both finite.

In Section 22 we shall show The conjecture of Birch and

Swinnerton-Dyer [27] predicts the following "class-number" formulae:

2h-2 ? = Card(W(A!F)) • h-l

p-4-

(

~ r(c!p);I(2~)m O
dc)=l

IT

O
In particular. one should have:

Card(Ul(A!H))

I

I

l

Card(UJ(A!F))2 •

ph!2)

72 22.

The rank of

sh,

A(p)

In this section we will apply the descent machinery we developed for (17.2 - 17.3) to obtain bounds on the ~-rank of

~-curves

A(p) • wh'

22.1.

We begin with the 2-descent when

Theorem 22.1.1.

Proof.

Since

A = A(p)

Assume that

A c ~2 x 2

\1

over

2

H

p - 7

with

JI

(mod 8) •

P _ 7

(mod 8).

Then

(14.2.1). Kummer theory gives an isomor-

phism:

,s c

A

*

*2 2 )

The map

A(H)/2A(H)~ (H /H

write

= 2R with R. A('ii).

P

H' = H( Icl. iii)

of

*

*2 2

(H /H

)

•

~ be described as follows.

Then

·,-i1

a

and

B

so that.

weJlI1lSt

detel:'lll1ne which ele.te,ot,"

, rf1(G8:I(f!i>J'2)"'\X*lr!2)2 eotD the cCllllpletioD

I

,

• I

A(P) • (a.B)

ll¥, (11.3.1)

P. A(H)

generates a biquadratic extension

R

H and we ~ choose

For

c' :' , "'.",i,'",

(H* /U*2)2

TI(U*/H*2)2. -v ..... v:

Let 0 v

Bo1 f

-

l 73 shows toot

ImA

when

vf2p.

If

v

I

2

0/20

v

then the image of

containing those elements of the form

(a,a)

A v with

is the subgroup of

aol+40

v

and

(Brumer and Kramer [4]).

Now assume

(a,a)

in

v

H

previous lemma and the fact that valuation at all places is

~

w of

not diViding

K

K,

is unrsmified over p

a

and

By the

v.

for all

A

a

have even

Since the class-number of'

K

we may write

a

=(_l)a(,r.:p)b

a =: with

a, b, c, d

onto

(H/H ) v

in

Z/2Z

This subgroup of

* *2 2 , where v has order

We nov

is any prime of

4, we see S2{A)

.a~lltld~18 a (lOPY of

(mod K*2 )

(_l)c(,r.:p)d

A(H)2 ..

study the

"..

H

(K*/K*2)2 restricts isomorphically dividing

has r&llk

0

0/20 i;y' (16.·2~S)

I:;

or and

p.

Since

lover

8

L

with

But

SO

Let 'L .. K(ll) p

"-.-

D

V

0/20

(11.

A .. A(p)·.

Im A ..

01;; 1

.

and

£!:.! '2

mod ! ! . }

..--

----------------_.~'-------

74 (22.2.2)

where

€=€

Vp

is the cyclotomic character defined in

section, we decompose

(13.1.3).

D only into eigenspaces for

characters of the full group Aut(L)

Notice that in this

Gal(L/K); later we will use the II

= Ga1(L/~) (lJcl.)

Theorem 22.2.3.

4

S (A) "D

A = A(p)

If

1T

n

Proof.

(13.1.2), Kummer theory gives an isomorphism:

Since

s

(22.2.4) 1 * * ~(Ga1(H/H),A ) " (H(v ) /H(V ) p) P

'1T

3 1 (po-T)

3 1

* (~) " (H(v )*/H(v ) p)

P

P

H(V )*/H(v )*P P P

where the right hand side denotes the sUb-space of acts via the character as follows:

for

P

£

A(H)

1?f = e:¥

p

of

(0-:;0/) ,

•

l!

the image of

(~1)-, "..

'(1 + ..P:jol'o 1(1 + 'It O:)p) v~, V"': - ',-'v.;"·--':

-

-

A may be described

Then R generates a so that

A(P)

,,'V' , .

'C';,·

H

=a

.,

~",:'1i.7.!iQ'"tO.'t:AAlluWbup "

'" .."

vip

A(H) •

p

.

<,"1) , ," ,"

€

H(v ) ; we may choose a

1:_ 22.N'.'lt ' #P >'\!ili'4brue ot'

* ,.'

The map

£'

.

on which p:

write P = 1TR with R

cyclic po-extension H(V )( %")

P

,";':

(

',.',

V is eQU!!:}. ,to the ,subgroup

A

F

75

If vtp

Proof. the points

with

R

,,=

Consequently

1m A

v

then multiplication by 1TR

A(P)

= P

vip

then

0

A(H) v

v

(¥)

= ZP [~ P ]

and

the finite group scheme

1T

H (~ ) v p

P. and

= (l-~)

v

H(Irr)

A

1T

in the Dort-Tate classification [15]. section.

0v' and

•

duction over the quadratic extension H (Irr,~ ) v p

is etale over

generated unramified extensions of

has valuation divisible by

= Av (A(HV1T ) ) = (O*/O*p) V V If

in

on A(p)

1T

The curve

of

A(p)

H; over the integers of

G _ rr, 7T

is isomorphic to the group Indeed A

1T

A

The computation of the image of

V

has good re-

is self-dual and has a rational

then follows from results of

Roberts [18] (see Appendix). (.lE::1.)

Now suppose previous lemma of the field Hence in

,,£

4

(L* /L*p)

(,,) = ~P • as the valuation of

C by an element

(22.2.1). Gal (till)

"

A

V

Let

"" 1

L . K(ll ) p

£

= (l-~)

is divisible by

(mod £).

= lIl(llp )

at all places

= K(llp ).' (~)

and def!;Qe thefl.n1te p.-group

C

as in

L • one has an exact sequence of

~

0

~ ~ ct(llt)

•

U/uP - . . C , et(t) (a) •.

Fori£Z let

By the

By the p-part of Lemma 22.2.5. ,,£ D

modules: D-

L

p

v.

is principal. we may represent ea:ch class

If U denotes the unit groJlllpf

(22.3.1)

for all

H(~p) • which is itself an unramified extension of

,,£ C ; since the ideal

.2 2.3.

is in the image of

.

76 Define the Bernoulli numbers by the power series expansion (m

-tL -e -1

The rational numbers

Lemma 22.3.4.

B

AssUlJle

u

of the form

it

B ~ 0 i

0 < i < p-l is even and

The eigenspace '.

be a

i < p - 1 •

are p-integral for

i

1) The eigenspace C(p-i) 2)

i ~I. B " -t I • i~O 1 1

C(i)

_

i

u = 1 + ae

(mod p).

Then

is trivial.

it t

has rank 1 ~ Z/pZ and is generated by a 1+1 (mod e' ).:!!tl.!!. a ~ 0 (mod e) .

The eigenspace decomposition of the global units is given by p-3 E& u/up (i) • where each even eigenspace appearing has rank 1

Since

i=2 i even (p-i)

is odd and greater than

1.

The latter space is annihilated by B ~ 0 (mod p). i Let

L

C

C e.- L'/L'p

This proves

U/UP(p-i) B i

.

C(p-i) = Ct(L) (p-i~ p

[11]. so is trivial by the assumption that

1) •

be the abelian p-extension of by

is trivial and

~UDIIler thllOrT.

Then

L

L which corresponds to the subgroup

C

maxh,' abelian p-eXt!!nsion of L ,,!>ich is raaitied

tuX'niSheli

Bence

~''''"..'".-•.; .oe,tCt.!Il)UQIIlOrphiSlll :'. :-.!':":: . ".'_.:: .-._> "_' c.-,·-:, :'.,-, -.:', :-0;: •. :.:_,~_-., . -_:: .:: . -.".:;.:

Ga3;CL~t»/L) {;t) . h

SinCe

Np)

j;rhie.l

byl)••

c~ntabls

claBs-tield theory turn:l.Bhe,

( (I ci(] 1e

-

77 Let (mod

u" 1

E.)

(mod

j

E. +1 ) with a

t

0

be a generator of this eigenspace, and write (mod

E.)

I claim that

be an unramified p-extension of

in

Ct(L) (p-i)

b

(Z/p~)

in

*

Hence

1)

If

would

= Gal(L/IIl)

(mod

Corollary 22.3.5.

L( Pill)

< p+l ; otherwise

L which would correspond to a non-trivial element

But for any

p

j

u = 1 + art

j

E:j+l )

= i , which completes the proof of

p" 7

(mod 8)

~

Bpi?

t

0

2) •

(mod p)

= (0)

S,,
2)

If

~.

p" 3

(mod 8)

and

B¥

t

0

S (A(p»

(mod p)

(¥) = D n(c(E!:1.4 ) 61 C(T)/ 3 1 \

Since, D

w

= O/wO

•

this follows from (22.2.3) and

(22.3.4). Actually, we can prove a slightly stronger result.

!t,.

!lRl? 1: 0

(mod p)

S,,(A(p» "

it BPf 1;0 (mod 2)

1!.P;; 3

(mod 8)

~

"

ettt)p(~) -(0) 1hlm "_'O:->:'<::C.

8 w(A(P»" 0/.0

'-'":::';C', :;C':.":':.-

. Proot•. It et(L) (p-i) ; P

p)

o.

i

B l! 0 ·(lllOd p) • a reaUl.t ot lIibetU.6J shows that i If· we assume further that 'et(L) (i) = (0) then . p

et(L)p(P-i) - C(p-i)

is even and

has rs.nlt 1 over

'lI/p'&

and is gener.ted by an element

a

of

78

22.4.

From the determination of the Selmer groups of

A(p)

at

n

and

2 ,

I we

we obtain the following results on the Ill-rank. Theorem 22.4.1.

Assume

1''' 7

(mod 8).

Then

n(A(p))

=a

MaZl

and

A(p)(H) = 0120

A(p}(F) = Z/2Z W(A(p))2 = (0)

W(A(p))n = Sn(A(p))

T, W,..'.1

Proof.

Since the map

8:A(H)2 -;::::::::- S2(A(p))

is an isomorphism (22.1.1) we

have:

= A(H) 2

A(H)/2A(H)Ga1(H/K)

The structure of

A(p)(F)

Theorem 22.4.2. n(A(p»

~ 1.

.!!.

and

Assume

A(p)(H)

1''' 3

A(p)(F) ; (0)

then folloWS from (14.2.1) and (16.2.5).

(mod 8)

n(A(p»

~ 1.

A(p)(F) '..

i'

A(p)(H) ..

~

The &SSllII!ptiOI1, !mplies thBt

by (17.3.2-3). .

':.;.:

If

structure of the gr~pli A(p)(F)

1 Cl(L)p(3r ) = (0)

~

W (A(p»" f!22!.

~

A(p) and

.. (0)

P~""'\PII.~· ....I~V

!!!!!!!.

,... \

79 I would expect that

n(A(p»

= 1

whenever

p" 3

(mod 8).

One can attempt to pro-

duce a non-trivial F-rational point us ing the methods of Heegner, Birch [3], and Mazur [14].

Recall the modular parametrization (20.3.2)

The prime

p

Write

= E. • E

(p)

splits in the imaginary quadratic field and let

R

corresponds to the modular data

Question 22.4.3.

Is

Ill( 1:2)

denote the rational point on {IC/Z:[ 1:2],

"split (Rl

(ker

E., ker

of class number Xsplit (p)

ElJ.

a non-trivial point in

A(p)(Fl ?

which

1 .

80

3.

2

Global mOdels. finE

23.1.

Recall that the discriminant ideal of

A(p)

is given by model

:A(p)/F) = (_p3).

Since this ideal is principal, global minimal models for

A(p) is

,rtainly exist when the class group of

F

is 12-divisible (Tate [28]).

,

E

We can ask ~

)r something stronger.

Question 23.1.1. 'iminant

Does

A(p)

have a model over the integers of

F

with dis-

t:. = _p3 1

Zagier has proposed' the following candidate.

Consider the equations

withc

~rwick [2], Birch [3]):

modl

e

that

m3 = j(A(p»

3.1.2) -pn

e first has a unique solution ; we can fix er

F

with

n

2

m

= j(A(p») in

by insisting that

- 1728

F. The second has two non-zero solutions in sign n

= (g,) p

Then the elliptic curve

A

e~tion:

y

2

..

3

:It . +

..JI!1!.,.

_4 _

It

-

2-'3

s

;,j (A(p»

DO

T

I I

23.2.

fined over

By (20.3.1) we have a non-trivial _p

F

and determined up to sign.

2 "O:XO(p ) - > A(p)

Assume that

A(p)

which is de-

has a global minimal

w be the associated Neron differential.

model of type (23.1.1) and let

Then

w

is also deter.mined up to sign. The local expansion of the differential

is the Fourier expansion of a cusp form

"o*(w)

fA(q) =

on

X (p2) O

r

a qn of weight n "0*(0;) generates HO(B n>1

with coefficients in

+

module over that

sign

T a

1

F

[22].

Since

= En~(BO) II III •

=1

a

1

F/III

; O.

Normalize the sign of

2

for

1 o.n )

"0

lim]

and

as a

w so

•

Question 23.2.2.

Is

a

1

=1

?

Are the Fourier coefficients of Is the modular form (20.2.1)1

Tr

at the cusp

g =

h1

fA

TrF/lllf A

all integers in

F?

equal to the theta-series

eB

defined in

82

24.

Compu tationa l examp les. 24.1.

There are

6

p > 3 where the class number of

primes

K = 1Il(,cp)

is

They are

equal toone .

p

= T

P

= II,

In these cases the curve

A(p)

(~)

=1

(~)

= -1

P

19, 43, 67, 163 is define d over

P

III

.

the follow ing minim al models

were provid ed by Tate. p

=7 c4 = 3'5'7 3

c6 = 3 ·7

2

= _7 3

l!.

-:n

= -33 '53

,

IX

Remar ks:

9R

The fundam ente.l ree.l period

of the l'Il!ron differ ential

Ol

.,,~x = .... ~

*

A(T)

on

and the integr al period lattic e

(21.1 .1.. 21.2.2 ) • are given by .

r(1£7l r (2fT}f (4{:t> ..

21r i

.r.::::r

W='1I!,O

trivie .l point of order The.cu rve

2

has coordi ll9.tes

A(7) is isomor phic to the modUl ar

to be the ident ity map in (20.1. 1).

W

\.

83

p = 11

y

2

+ y

= x

3

- x

2

- 7x + 2- 5

5 c4 = 2 '11

3 2 c6 = -2 '7'11

~

Remarks: differential

The fundamental real period and integral period lattice of the Neron

w = 2y~1 are given by (21.1.1., 21.3.2) •

~=

w Since

Be- =

1 30

In fact, the rank is

(x,y)

= (4,5)

= _11 3

~

(2~i)2

~. V- l

=

0 1

r(1/11)r(3/11)r(4/11)r(5/11)r(g/11)

(mod 11) , the curve

A(11)

has

III rank ~ 1

(22.4.2).

A(ll)(lIl) = Z •. The point with coordinates

and

is a generator.

For a discussion of the modular parametrization as weli as the isomorphism

Xnon_split(li)

y

2

= A(li),

3

.

. ....•. .. ... ,".;-::-<.-. -.:-.,

c4 = 2 '3'.1:9

c6

2 = -23'33'19

A = _193

J =

_:15'33

X (l1) ..... A(ll) split

see Ligozat [13].

J:~-~

+y=x·-2'l!lX'+, . . 5

~:XO(121) -+

-----------

----.-,

1IIIIIIII

---~.

84

p =

43

y

2

+ y

= x3

2 - 2 '5·43 x

2

+

3'7·43 -1 4

6 c4 = 2 '3'5·43

p =

p

c6

= -2 3 '3 4 '7·432

f1

=

j

= _218 '33'5 3

c4

= 25'3'5'11'67

c6

= -2 3 '33 '7'31'67 2

f1

= _67 3

j

= _215 '33'53'113

c4

= 26'3'5'23'29'163

_43 3

67

calcul

= 163 Am:

'.i

c6 • -23'33'7'11'19'127'1632 A --1633

":2",~.j$);~3. 233•293 '

period attic'. of tihe .iZ09D,4trferexlti,rJ,.

DOD-tx the n:

~

85

24.2.

P

= 19

(x,y)

= (4,1)

p

= 43

(x,y)

= (17,0)

p

= 67

(x,y)

= (201/4,63/8)

P

= 163

(x,y)

= (850,68)

We will consider 2 cases when K = 1Il(,.cp) has class number 3

I

p

= 23

(.£) p

=1

59

(.£) p

= -1

1 f I

P =

I I

I I

In these cases the curve

A(p)

is defined over the cubic field

F

= lIl(j).

For a

calculation of the relevant cubic moduli, see Berwick [ 2].

I

A minimal model for

p

= 23

F

= 1Il(,,)

A(23)

where

,,3 _ " _ 1 =

°.

of type (23.1.1) is given by:

The in

86

p = 59

F A minimal model for

"4

= 1Il(,,) A(59)

= ,,4

where

"

3

2

- 2" - 1 = 0 •

of type (23.1.1) is given by

• 25 (,,-1)(2,,-1) 59

_ ,,623(4<>2_6<>_3)(2,,2_2,,_1)(3,,_2)

"6 '"

(3<>-4)

= -59

•

2 59

3

SCh~

-

Since B44

=0

(mod 59) , Theorem 22.3.5 does not applyl

(22.4.2) to conclude that A(53)(F) order.

= i3.

n(A(53»

~

1.

The point with coordinates

But we can apply

In fact, the Ill-rank is (x,y)

r

1

= (5,,2+ 6 ,_2,,2+2"+5)

and has infinite such I

tb

!2!

'desce pr

ri

T P

of

I

mU!:It

(c l(

,.... I

I

I 25.

The cohomologY of the Fermat group scheme

Appendix:

B. Mazur

by

Let

25.1-

be a prime number

p

u = 1 - ~p

A = ring of integers in

lIl(u) ; S=SpecA

B = ring of integers in

1Il(';;) ; T = Spec B

Proposition 25.1.1. schemes over

T

~

Up to isomorphism in the category of finite flat group

there is a unique diagram

Zip

such that

j

0

the morphisms

....!- c ....L,.

i

brings the section

i

and

j

"1 1t

of

].I

p

'li/p

to the section

"Cpu

.£!'.)

lJ

p

,

are isomorphisms of group schemes when restricted to the

base T[l/p] •. and either one of the following conditions hold· a)

Neither

.!!.2!:. j

i

are 1iI0!1l!ll1?hisms over T.

The above diagram is Self~duaqdtbrellpec:ttoCartie1"duality"'

Proof. . A diagram of the above sOrl is '''rigid"

and therefore an elementary

. desCel1;1;,(/UlQ aPPJ'oxlm,stion) argllm~t willsllow.exis.tence and uniqueness over T

.~~, we "."

. . . ".

...

>..

.-

have' demonstrated existence an!i .'uniquenesS, over·. the cOlllPlete' iocal base ....

_...

..

.

..

,-.- .

.-'

..

-

- . ··.i··~·:

~-";'-

Tl":' >Not... that. C/Tp'is an Oort-Tate lltouP scheme", G"",b of Bp . such that

a-b- p.

must have, ord(a)and ord(b) ,that .ord(h) .• 1).

Since

Since

~ere:,.a,b are elements

Ga,l;' has its sections rationsl over Tp

both dinsible by. p ... 1· (ord is ,nor:msliZed so

ord(p). 2(p-l)

this leaves

3

possibilities for

.';~d~{~~i~?~Q~li,H~"~ .~,~ell~~'f
we

Let

j

"upper index" sub-group of the units in

U(j) =

V(j)

Proposition 25.1.3. with the sUbgroup

f!:22!.

VIp)

The map ~

A

p

and

fB*)(j)

"",,1'1

p

B

p

p

denote the

respe~tively.

with

V(j)

= (B*)(j)B*P/B*P P

P

c...L-,. ~

P

P

induces an isomorphism of

1

VIol = H (T .~ )

P P

where

j

= 2p

putationgives that the discriminant of

To compute the global cohomology of Lemma 25.1.4.

_ ord(disc C) p - 1

CiT P

is

identi-

But an elementary com-

(,r,r)p(p-l)

C we need:

The following square is Cartesian:

1'joot.BtIlDd4rd.

Either by cOlll!P&r1soll-otgl~bl~ ;~o' :local 8.1raC1~-·84Mt'dI~Q"8~

a _-'!:trio, Sl'pll$llt intel'j>re1;ing .. <>;. .\ ;'. "_'''''.. ,,::,.!_:.' ·:c· .. · .'.' .. :.. .> . . ' .... ~:'f.

Let

(A*)(j)A*P/A*P P P P

Theorem 1 of Roberts' Thesis ([ 4); see also. [3. Prop. 9.3J

fiesI1(T 'C) p

:/:~_.

();*)\J)

be a non-negative integer and let

':('~.

II

_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _iiiiiiiiiiiiiiiii'''iii'''iiTiii''ii'·'ii'.'iiiiiiiiiSFiiiiiiRiiii

,l-.. .

...

89

~(T p .C)

in ~(T .~ ). p

p

We define

ffl(s.c) = ~(T.C)G .

(25.2.1)

Theorem 25.2.2.

If P is odd p+1

and ,," 1

Proof.

mod"

2

)

From (25.1.4) one obtains that the square

~(s.c) - - ihs .c)

1

i1(s.~ p ) is Cartesian. where

i1 (S ._)

=

~ (T ._) G.

(

i1(Sp .~p ) By Kummer theory

Taking invariants gives

as

pia

pr1llla to

laj • S1llIilarly V(O)a. tI(ol 'ilJldV(p)a • tI(Pl).

g1ves tbe theorem. ',:', ,,", ~,.

Co~ll!l7 25.2.3. l! p 1s

'

..

an odd.. rml1lH"~!t! 1t

'p

=1 .(lIIQd Ii)

1t.p il3

(mod 4)

This

90

!1:22!.

~(S'~J

tion,

be the units of

is non-t rivial for

E ~p

.

'+1

(mod ,,2

i > p+l

-

lIl( ~p)

The eigens pace of

is one-di mensi onal; since

u " 1 + a,,2

such

= U/UP . i

the charac t er

U/UP( i)

U = A*

Let

)

where

a

t

U/U P

on which

C!(A)p = (0) Gal(lIl(~p)/lIl)

i=1, 2,4, 6, ... , p-3.

by as sumpacts via

that

,

grouI

In each case

p

is regula r it can be genera ted by a unit

to ti..-:

0

(mod" )

T ,

is theref ore the dimen sion of

2

Since

(compa re (22.3 .4».

The number of

genouS

ffl(s,c) .

that

25.3.

We shall apply the above compu tation to retrie ve (part of) a result of

Faddee v [1 J•

Let

K

= lIl( ~p ) = lIl(,,)

plex multip licatio n by

K.

over

",

and let

J /K

be an abelia n variet y with com-

posses sing a non-t rivial point of order

p

is an odd regula r prime.

(ii)

J

achiev es good reduct ion everyw here over

Examp les of abelia n variet ies with multi plicat ion by p" 1

(mod 3)

Define the 'r-flelm e:r numbe r of. J

Theorem 2'1' ~.l.

"

p

1Ihere

Over

in

.1:

as

P

satisfY ing p

(ii)

are

where the

Th

has a "tame" quotie nt (Gross "';Roh rlich

by

Under the above hYpOt heses:

. -t:

Proof.

ration al

T.

and, more gener ally, for those primes

Jacobi an of the Ferma t curve of expone nt

[2] )•

"

Suppos e furthe r that:

(i)

known for

quence

T' 'We haVe 'the follow ing short exact

2

3

4.

a

91

mp-

that

is an isogeny of abelian schemes) and therefore

ia

group scheme over

1T

T

of order

non-trivial point of order to the constant group f

T ,

ker

1T

_

1T

p

= deg

n.

is a finite flat

11"

By the assUMption that

it follows that

,

ker

~

-ker

J(K)

is isomorphic over

is either isomorphic to

ker n

T[l/p]

By the discussion in (26.1) we may conclude that, over

7ljp

Zip,

or

C,

~

p

.

Since

ker

1T

genous to its dual, it cannot be either etale or of multiplicative type. that

has a

is iso-

It follows

C , and the exact se~uence (26.3.2) gives the se-

is isomorphic to

quence: o

-T

in flat cohomology. as

p

J(T)/1rJ(T)

The

Jil-(T,C) -

-T

G.

-T

0

invariants of this se~uence remain exact,

G = Gal(T!S)

is prime to the order of

W(J!T) n

A s:jmilar argument shows that

J(T)!nJ(T)G

= J(K)!~J(K)

W(J!T)G = ~

The asserted formula then follows from

W

(J!K)

~

•

d~p:D: J(K)!~J(K)

=1

+

d~(K)&AK

.

References:

k R. in "Iilvariants of divisor classes··tor the ·c:urves x' (l-X) " y fieid. Tritdy Math.:- (in Russian) Inst. Steklov 64 ....adic etelotomic

Faddeev,. D.

. e.n

1(;

(1961), 284-293. 2.

Gross, B. H. and Rohrlieh, D.E. SOllle res1lJ.tJl anthe·M!iI'4eJ.J.,-wehgroup of the il"aeobian af the Fermat curve. !nv. Math. 44 q.9'(8) ,.201-224.

3. Mazur~ B. and Roberts, L. -201-234. Roberts, L. On the flat eohomolollY of finite group scihemes~ TheSis. (1968).

Harvard

26. 1.

Arthau d, N.

Prepr int.

S}

23·

Berwic k, W. E. H. Modul ar invari ants expre ssible in terms of quadra tic and cubic irrati onali ties. Proc. London Math. Soc. (2)

3.

Birch, B. J. Dioph antive analys is and modul ar functi ons. quium on Algeb raic Geome try (1968) , 35-42.

4.

Brume r, A. and Krame r, K.

(1977) , 715-74 3.

6.

22.

On Birch and SWinn erton-D yer's conjec ture for ellipt ic curves with

28 (1927) , 53-69.

5.

S,

Biblio graphy .

comple x multip licatio n II. 2.

21.

The rank of ellipt ic curves .

Duke Math. J. (4) 44,

On Epste in's zeta-f unctio n.

Crelle J. 227 (1967 ),

Chowl a, S. ana Selber g, A. 86-ll0 .

24.

~}

25·

:1

Proc. Bombay Collo-

26. 27.

s

28.

T

29.

~

30.

[

31.

,~

32.

1

33.

I

Delign e, P. Courbe s ellipt iques : formu laire. Modul ar functi ons of one variable (Antwe rp IV). Lectur e Notes in Math. 476 (1975

), 53-73.

7.

Delign e, P. Valeu rs de foncti ons L et p~riodes d' int~gr ales. in Pure Math 33 (1979) , part 2, 313-34 6.

8.

Deurin g, M. Die zetai'U nktion einer algebr aische n Kurve von Gesch lechte Eins. I - IV. Gott. Nach. (1953, 1955-1 957).

9.

Gross, B. and Kobli tz, N.

109 (1979 ), 569-58 1.

Gauss sums and the p-adic r-func tion.

10.

Gross , B.

ll.

~,

12.

Lang, S.

13:"

Li~Z::r.G. {:=~~~t~~

14.

S.

1faZ1U','~;

Ill-curv es and p-adic L-fun ctions . Cyclot omic fields . Ellip tic

In prepa ration .

lIMtBo n-WeB le;y. {:l973h..

ModUlar. curves and the

~ .',"',

Annals Math.

Spr1ll ger-Ve rlag (1978) .

~f'wlctlons.

, ;v,,~,,' ,(~~).~~~(l6~,>

Froc. Symp.

<

'.'

,<'

,\,;,o.c;,'\ ',"

. .", Els~titin~ ~. <

,

,~~~

;

"

I't ,

";,;'~';: ":':

34. 'I

";:i::·'''~>.~~:;:,,'r;i;~~:;}" ...';' :,' ~:*i1; .,'~:( ',":

15.0 0rt, F. and TateJ •. Group schemell ot. Pl"tl!ilil. "

'.' ,ADil.

'scr,';Urs.

~ ~ ~ ,,:,',.~.'<~.<...et~,,~,_: ~~P,J~~":~'(:~;'1·6~.', 'c. '~.;n,'.;i.;,·.·'onJ·,.i."~:~, :.".!,.,.,:,,,:i ~, ."~.,.~';~l>'~~;~,:,~;'O'" .' _~..;,o+-::, 16.'.

:l'"'ns.",'t_'

l'

IT.

" .........,

" i , " Sl(;:';";'<,"

.:

"

Rober t, G.

18.

77"

.~

t"

-

93

21.

Serre, J.-P. and Tate, J.

Good reduction of abelian varieties.

Annals Math.

88 (1968), 492-517.

es

22.

Shimura, G. Introduction to the arithmetic theo,,' of automorphic functions. Publ. Math. Soc. Japan 11 (1971).

23.

Shimura, G. On elliptic curves with complex multiplication as factors of the Jacobians of modular function fields. Nagoya Math. J. 43 (1971), 199-

with

208.

md )

.

24. :0110-

Shimura, G. field.

) 44,

Shimura, G. On the zeta-function of an abelian variety with complex multiplication. Annals Math. 94 (1971), 504-533.

967) ,

26.

Shimura,

27.

SWinnerton-Dyer, H. P. F. The conjecture of Birch and Swinnerton-Dyer, and of Tate. Proc. Driebergen Conference on local fields. Springer (1967),

G.

On the periods of modular forms.

28.

Tate, J. Algorithm for determining the type of a singular fibre in an elliptic pencil. Modular forms of one variable (Antwerp IV). Lecture Notes in Math. 476 (1975), 33-52.

29.

Tate, J.

The arithmetic of elliptic curves.

30.

Tate, J.

Endomorphisms of abelian varieties over finite fields.

np.

23 (1974), 179-206•

Inv. Math

Inv. Math. 2,

(1966),134-144.

lath.

31.

Tate, J. Local constants. Proc. Symposium on algebraic number fields. Academic Press (1977). 89-133.

32.

Weil. A.

Adl!les and algebraic groups.

Study

33.

Princeton:

Institute for Advanced

(1961) •

)lell. A. EJ.:I.!lltic functions acocordillgto Etsenstein~4_Ktonecker; S!»,inger-Verl&8 (1976). " , -_. .

34;' Le

229 (l977) , 211-221.

Math. Ann.

132-157.

vari-

.nsa

On the factors of the Jacobian variety of a modular function Jour. Math. Soc. Japan 25 (1973), 523-544.

Wei1, A~

gf!n.

.

..

.

BestiJlllllune lliricb1stsc~ Reiben dureb Fwlltticmal gleichunMath. Ann. 168 (1967). 149-156.':

fiber' die

0

00

0

0

'

0

'

,

, "

•

~---.-_._-----~

27.

Index.

3, 26, 32 6, 24 1, 35, 41, 67 37 A

+ , A _

q

13, 41

q

B .. ll-/Ill(A)

45, 57, 60

C,

73, 75, 76

C(i)

C(p) ,

..

C(p)

38

cl(K) .. ideal class-group of K

12

complex multiplication

It 12,

20

73, 74 descended curve

1, 29

elliptic curve .F .. Ill{J) ;. field of /llQduli h .. [H:K] .. clafIs:-number of H ..

Hilbert

K

Class<-field of K

Iii .. idMes of

Ii

23

1sog~

K .. imaginary quadratic field

1, 29, 34

95

L(A/F.s)

= L-series

of A

19. 22. 58 58. 60

minimal model

14. 80

modular form

64. 65. 81 j(A)

1.

modular parametrization

66

modular invariant

~(A) •

'1!(A) • n(A)

o = ring

of integers of K

5. 23

1. 29. 34

35. 40. 42 q-expansion

65. 66. 81

Ill-curve

1. 32

Ill-rank

n(A)

49. 72

R = En~(B)

S (AIR) = Sellner group

53

S,,(A)

55. 72. 77

"

sign(a)

= faT

UJ(A/H) = Tate-Shafarevitch group,

40. 60 53. 71

(

,

56. 78 47. 48. 49. 57 17 Weierstrass model

4

64. 66

17. 20